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Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaovassg 5601* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))       ((φ (A 𝑆 B 𝑆 𝐶 𝑆)) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))

Theoremcaovassd 5602* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)       (φ → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))

Theoremcaovass 5603* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
A V    &   B V    &   𝐶 V    &   ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))       ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶))

Theoremcaovcang 5604* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))       ((φ (A 𝑇 B 𝑆 𝐶 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))

Theoremcaovcand 5605* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))    &   (φA 𝑇)    &   (φB 𝑆)    &   (φ𝐶 𝑆)       (φ → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))

Theoremcaovcanrd 5606* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))    &   (φA 𝑇)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   (φA 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))       (φ → ((B𝐹A) = (𝐶𝐹A) ↔ B = 𝐶))

Theoremcaovcan 5607* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
𝐶 V    &   ((x 𝑆 y 𝑆) → ((x𝐹y) = (x𝐹z) → y = z))       ((A 𝑆 B 𝑆) → ((A𝐹B) = (A𝐹𝐶) → B = 𝐶))

Theoremcaovordig 5608* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y → (z𝐹x)𝑅(z𝐹y)))       ((φ (A 𝑆 B 𝑆 𝐶 𝑆)) → (A𝑅B → (𝐶𝐹A)𝑅(𝐶𝐹B)))

Theoremcaovordid 5609* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y → (z𝐹x)𝑅(z𝐹y)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)       (φ → (A𝑅B → (𝐶𝐹A)𝑅(𝐶𝐹B)))

Theoremcaovordg 5610* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))       ((φ (A 𝑆 B 𝑆 𝐶 𝑆)) → (A𝑅B ↔ (𝐶𝐹A)𝑅(𝐶𝐹B)))

Theoremcaovordd 5611* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)       (φ → (A𝑅B ↔ (𝐶𝐹A)𝑅(𝐶𝐹B)))

Theoremcaovord2d 5612* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))       (φ → (A𝑅B ↔ (A𝐹𝐶)𝑅(B𝐹𝐶)))

Theoremcaovord3d 5613* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   (φ𝐷 𝑆)       (φ → ((A𝐹B) = (𝐶𝐹𝐷) → (A𝑅𝐶𝐷𝑅B)))

Theoremcaovord 5614* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
A V    &   B V    &   (z 𝑆 → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))       (𝐶 𝑆 → (A𝑅B ↔ (𝐶𝐹A)𝑅(𝐶𝐹B)))

Theoremcaovord2 5615* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
A V    &   B V    &   (z 𝑆 → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))    &   𝐶 V    &   (x𝐹y) = (y𝐹x)       (𝐶 𝑆 → (A𝑅B ↔ (A𝐹𝐶)𝑅(B𝐹𝐶)))

Theoremcaovord3 5616* Ordering law. (Contributed by NM, 29-Feb-1996.)
A V    &   B V    &   (z 𝑆 → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))    &   𝐶 V    &   (x𝐹y) = (y𝐹x)    &   𝐷 V       (((B 𝑆 𝐶 𝑆) (A𝐹B) = (𝐶𝐹𝐷)) → (A𝑅𝐶𝐷𝑅B))

Theoremcaovdig 5617* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
((φ (x 𝐾 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐻(x𝐺z)))       ((φ (A 𝐾 B 𝑆 𝐶 𝑆)) → (A𝐺(B𝐹𝐶)) = ((A𝐺B)𝐻(A𝐺𝐶)))

Theoremcaovdid 5618* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝐾 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐻(x𝐺z)))    &   (φA 𝐾)    &   (φB 𝑆)    &   (φ𝐶 𝑆)       (φ → (A𝐺(B𝐹𝐶)) = ((A𝐺B)𝐻(A𝐺𝐶)))

Theoremcaovdir2d 5619* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))       (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐹(B𝐺𝐶)))

Theoremcaovdirg 5620* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))       ((φ (A 𝑆 B 𝑆 𝐶 𝐾)) → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))

Theoremcaovdird 5621* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝐾)       (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))

Theoremcaovdi 5622* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
A V    &   B V    &   𝐶 V    &   (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z))       (A𝐺(B𝐹𝐶)) = ((A𝐺B)𝐹(A𝐺𝐶))

Theoremcaov32d 5623* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))       (φ → ((A𝐹B)𝐹𝐶) = ((A𝐹𝐶)𝐹B))

Theoremcaov12d 5624* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))       (φ → (A𝐹(B𝐹𝐶)) = (B𝐹(A𝐹𝐶)))

Theoremcaov31d 5625* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))       (φ → ((A𝐹B)𝐹𝐶) = ((𝐶𝐹B)𝐹A))

Theoremcaov13d 5626* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))       (φ → (A𝐹(B𝐹𝐶)) = (𝐶𝐹(B𝐹A)))

Theoremcaov4d 5627* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   (φ𝐷 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)       (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(B𝐹𝐷)))

Theoremcaov411d 5628* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   (φ𝐷 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)       (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹B)𝐹(A𝐹𝐷)))

Theoremcaov42d 5629* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   (φ𝐷 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)       (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(𝐷𝐹B)))

Theoremcaov32 5630* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
A V    &   B V    &   𝐶 V    &   (x𝐹y) = (y𝐹x)    &   ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))       ((A𝐹B)𝐹𝐶) = ((A𝐹𝐶)𝐹B)

Theoremcaov12 5631* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
A V    &   B V    &   𝐶 V    &   (x𝐹y) = (y𝐹x)    &   ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))       (A𝐹(B𝐹𝐶)) = (B𝐹(A𝐹𝐶))

Theoremcaov31 5632* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
A V    &   B V    &   𝐶 V    &   (x𝐹y) = (y𝐹x)    &   ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))       ((A𝐹B)𝐹𝐶) = ((𝐶𝐹B)𝐹A)

Theoremcaov13 5633* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
A V    &   B V    &   𝐶 V    &   (x𝐹y) = (y𝐹x)    &   ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))       (A𝐹(B𝐹𝐶)) = (𝐶𝐹(B𝐹A))

Theoremcaovdilemd 5634* Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z)))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z)))    &   ((φ (x 𝑆 y 𝑆)) → (x𝐺y) 𝑆)    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   (φ𝐷 𝑆)    &   (φ𝐻 𝑆)       (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))

Theoremcaovlem2d 5635* Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z)))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z)))    &   ((φ (x 𝑆 y 𝑆)) → (x𝐺y) 𝑆)    &   (φA 𝑆)    &   (φB 𝑆)    &   (φ𝐶 𝑆)    &   (φ𝐷 𝑆)    &   (φ𝐻 𝑆)    &   (φ𝑅 𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)       (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = ((A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))

Theoremcaovimo 5636* Uniqueness of inverse element in commutative, associative operation with identity. The identity element is B. (Contributed by Jim Kingdon, 18-Sep-2019.)
B 𝑆    &   ((x 𝑆 y 𝑆) → (x𝐹y) = (y𝐹x))    &   ((x 𝑆 y 𝑆 z 𝑆) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))    &   (x 𝑆 → (x𝐹B) = x)       (A 𝑆∃*w(w 𝑆 (A𝐹w) = B))

Theoremgrprinvlem 5637* Lemma for grprinvd 5638. (Contributed by NM, 9-Aug-2013.)
((φ x B y B) → (x + y) B)    &   (φ𝑂 B)    &   ((φ x B) → (𝑂 + x) = x)    &   ((φ (x B y B z B)) → ((x + y) + z) = (x + (y + z)))    &   ((φ x B) → y B (y + x) = 𝑂)    &   ((φ ψ) → 𝑋 B)    &   ((φ ψ) → (𝑋 + 𝑋) = 𝑋)       ((φ ψ) → 𝑋 = 𝑂)

Theoremgrprinvd 5638* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
((φ x B y B) → (x + y) B)    &   (φ𝑂 B)    &   ((φ x B) → (𝑂 + x) = x)    &   ((φ (x B y B z B)) → ((x + y) + z) = (x + (y + z)))    &   ((φ x B) → y B (y + x) = 𝑂)    &   ((φ ψ) → 𝑋 B)    &   ((φ ψ) → 𝑁 B)    &   ((φ ψ) → (𝑁 + 𝑋) = 𝑂)       ((φ ψ) → (𝑋 + 𝑁) = 𝑂)

Theoremgrpridd 5639* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
((φ x B y B) → (x + y) B)    &   (φ𝑂 B)    &   ((φ x B) → (𝑂 + x) = x)    &   ((φ (x B y B z B)) → ((x + y) + z) = (x + (y + z)))    &   ((φ x B) → y B (y + x) = 𝑂)       ((φ x B) → (x + 𝑂) = x)

2.6.11  "Maps to" notation

Theoremelmpt2cl 5640* If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (x A, y B𝐶)       (𝑋 (𝑆𝐹𝑇) → (𝑆 A 𝑇 B))

Theoremelmpt2cl1 5641* If a two-parameter class is not empty, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (x A, y B𝐶)       (𝑋 (𝑆𝐹𝑇) → 𝑆 A)

Theoremelmpt2cl2 5642* If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (x A, y B𝐶)       (𝑋 (𝑆𝐹𝑇) → 𝑇 B)

Theoremelovmpt2 5643* Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐷 = (𝑎 A, 𝑏 B𝐶)    &   𝐶 V    &   ((𝑎 = 𝑋 𝑏 = 𝑌) → 𝐶 = 𝐸)       (𝐹 (𝑋𝐷𝑌) ↔ (𝑋 A 𝑌 B 𝐹 𝐸))

Theoremf1ocnvd 5644* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (x A𝐶)    &   ((φ x A) → 𝐶 𝑊)    &   ((φ y B) → 𝐷 𝑋)    &   (φ → ((x A y = 𝐶) ↔ (y B x = 𝐷)))       (φ → (𝐹:A1-1-ontoB 𝐹 = (y B𝐷)))

Theoremf1od 5645* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (x A𝐶)    &   ((φ x A) → 𝐶 𝑊)    &   ((φ y B) → 𝐷 𝑋)    &   (φ → ((x A y = 𝐶) ↔ (y B x = 𝐷)))       (φ𝐹:A1-1-ontoB)

Theoremf1ocnv2d 5646* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (x A𝐶)    &   ((φ x A) → 𝐶 B)    &   ((φ y B) → 𝐷 A)    &   ((φ (x A y B)) → (x = 𝐷y = 𝐶))       (φ → (𝐹:A1-1-ontoB 𝐹 = (y B𝐷)))

Theoremf1o2d 5647* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (x A𝐶)    &   ((φ x A) → 𝐶 B)    &   ((φ y B) → 𝐷 A)    &   ((φ (x A y B)) → (x = 𝐷y = 𝐶))       (φ𝐹:A1-1-ontoB)

Theoremf1opw2 5648* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5649 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
(φ𝐹:A1-1-ontoB)    &   (φ → (𝐹𝑎) V)    &   (φ → (𝐹𝑏) V)       (φ → (𝑏 𝒫 A ↦ (𝐹𝑏)):𝒫 A1-1-onto→𝒫 B)

Theoremf1opw 5649* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝐹:A1-1-ontoB → (𝑏 𝒫 A ↦ (𝐹𝑏)):𝒫 A1-1-onto→𝒫 B)

Theoremsuppssfv 5650* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(φ → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   (φ → (𝐹𝑌) = 𝑍)    &   ((φ x 𝐷) → A 𝑉)       (φ → ((x 𝐷 ↦ (𝐹A)) “ (V ∖ {𝑍})) ⊆ 𝐿)

Theoremsuppssov1 5651* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(φ → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)    &   ((φ x 𝐷) → A 𝑉)    &   ((φ x 𝐷) → B 𝑅)       (φ → ((x 𝐷 ↦ (A𝑂B)) “ (V ∖ {𝑍})) ⊆ 𝐿)

2.6.12  Function operation

Syntaxcof 5652 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅

Syntaxcofr 5653 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅

Definitiondf-of 5654* Define the function operation map. The definition is designed so that if 𝑅 is a binary operation, then 𝑓 𝑅 is the analogous operation on functions which corresponds to applying 𝑅 pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))

Definitiondf-ofr 5655* Define the function relation map. The definition is designed so that if 𝑅 is a binary relation, then 𝑓 𝑅 is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑟 𝑅 = {⟨f, g⟩ ∣ x (dom f ∩ dom g)(fx)𝑅(gx)}

Theoremofeq 5656 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Theoremofreq 5657 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)

Theoremofexg 5658 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(A 𝑉 → ( ∘𝑓 𝑅A) V)

Theoremnfof 5659* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
x𝑅       x𝑓 𝑅

Theoremnfofr 5660* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
x𝑅       x𝑟 𝑅

Theoremoffval 5661* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝑆    &   ((φ x A) → (𝐹x) = 𝐶)    &   ((φ x B) → (𝐺x) = 𝐷)       (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ (𝐶𝑅𝐷)))

Theoremofrfval 5662* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝑆    &   ((φ x A) → (𝐹x) = 𝐶)    &   ((φ x B) → (𝐺x) = 𝐷)       (φ → (𝐹𝑟 𝑅𝐺x 𝑆 𝐶𝑅𝐷))

Theoremfnofval 5663 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝑆    &   ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)    &   ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)    &   (φ𝑅 Fn (𝑈 × 𝑉))    &   (φ𝐶 𝑈)    &   (φ𝐷 𝑉)       ((φ 𝑋 𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Theoremofrval 5664 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝑆    &   ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)    &   ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)       ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝐶𝑅𝐷)

Theoremofmresval 5665 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(φ𝐹 A)    &   (φ𝐺 B)       (φ → (𝐹( ∘𝑓 𝑅 ↾ (A × B))𝐺) = (𝐹𝑓 𝑅𝐺))

Theoremoff 5666* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
((φ (x 𝑆 y 𝑇)) → (x𝑅y) 𝑈)    &   (φ𝐹:A𝑆)    &   (φ𝐺:B𝑇)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝐶       (φ → (𝐹𝑓 𝑅𝐺):𝐶𝑈)

Theoremofres 5667 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φA 𝑉)    &   (φB 𝑊)    &   (AB) = 𝐶       (φ → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))

Theoremoffval2 5668* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
(φA 𝑉)    &   ((φ x A) → B 𝑊)    &   ((φ x A) → 𝐶 𝑋)    &   (φ𝐹 = (x AB))    &   (φ𝐺 = (x A𝐶))       (φ → (𝐹𝑓 𝑅𝐺) = (x A ↦ (B𝑅𝐶)))

Theoremofrfval2 5669* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
(φA 𝑉)    &   ((φ x A) → B 𝑊)    &   ((φ x A) → 𝐶 𝑋)    &   (φ𝐹 = (x AB))    &   (φ𝐺 = (x A𝐶))       (φ → (𝐹𝑟 𝑅𝐺x A B𝑅𝐶))

Theoremsuppssof1 5670* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(φ → (A “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)    &   (φA:𝐷𝑉)    &   (φB:𝐷𝑅)    &   (φ𝐷 𝑊)       (φ → ((A𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿)

Theoremofco 5671 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn B)    &   (φ𝐻:𝐷𝐶)    &   (φA 𝑉)    &   (φB 𝑊)    &   (φ𝐷 𝑋)    &   (AB) = 𝐶       (φ → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Theoremoffveqb 5672* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(φA 𝑉)    &   (φ𝐹 Fn A)    &   (φ𝐺 Fn A)    &   (φ𝐻 Fn A)    &   ((φ x A) → (𝐹x) = B)    &   ((φ x A) → (𝐺x) = 𝐶)       (φ → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ x A (𝐻x) = (B𝑅𝐶)))

Theoremofc12 5673 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φA 𝑉)    &   (φB 𝑊)    &   (φ𝐶 𝑋)       (φ → ((A × {B}) ∘𝑓 𝑅(A × {𝐶})) = (A × {(B𝑅𝐶)}))

Theoremcaofref 5674* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φA 𝑉)    &   (φ𝐹:A𝑆)    &   ((φ x 𝑆) → x𝑅x)       (φ𝐹𝑟 𝑅𝐹)

Theoremcaofinvl 5675* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(φA 𝑉)    &   (φ𝐹:A𝑆)    &   (φB 𝑊)    &   (φ𝑁:𝑆𝑆)    &   (φ𝐺 = (v A ↦ (𝑁‘(𝐹v))))    &   ((φ x 𝑆) → ((𝑁x)𝑅x) = B)       (φ → (𝐺𝑓 𝑅𝐹) = (A × {B}))

Theoremcaofcom 5676* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(φA 𝑉)    &   (φ𝐹:A𝑆)    &   (φ𝐺:A𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝑅y) = (y𝑅x))       (φ → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))

Theoremcaofrss 5677* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φA 𝑉)    &   (φ𝐹:A𝑆)    &   (φ𝐺:A𝑆)    &   ((φ (x 𝑆 y 𝑆)) → (x𝑅yx𝑇y))       (φ → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))

Theoremcaoftrn 5678* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(φA 𝑉)    &   (φ𝐹:A𝑆)    &   (φ𝐺:A𝑆)    &   (φ𝐻:A𝑆)    &   ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝑅y y𝑇z) → x𝑈z))       (φ → ((𝐹𝑟 𝑅𝐺 𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))

2.6.13  Functions (continued)

TheoremresfunexgALT 5679 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5325 but requires ax-pow 3918 and ax-un 4136. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun A B 𝐶) → (AB) V)

Theoremcofunexg 5680 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun A B 𝐶) → (AB) V)

Theoremcofunex2g 5681 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((A 𝑉 Fun B) → (AB) V)

TheoremfnexALT 5682 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 4926. This version of fnex 5326 uses ax-pow 3918 and ax-un 4136, whereas fnex 5326 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn A A B) → 𝐹 V)

Theoremfunrnex 5683 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5327. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹 B → (Fun 𝐹 → ran 𝐹 V))

Theoremfornex 5684 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(A 𝐶 → (𝐹:AontoBB V))

Theoremf1dmex 5685 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
((𝐹:A1-1B B 𝐶) → A V)

Theoremabrexex 5686* Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B(x). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5329, funex 5327, fnex 5326, resfunexg 5325, and funimaexg 4926. See also abrexex2 5693. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
A V       {yx A y = B} V

Theoremabrexexg 5687* Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. The antecedent assures us that A is a set. (Contributed by NM, 3-Nov-2003.)
(A 𝑉 → {yx A y = B} V)

Theoremiunexg 5688* The existence of an indexed union. x is normally a free-variable parameter in B. (Contributed by NM, 23-Mar-2006.)
((A 𝑉 x A B 𝑊) → x A B V)

Theoremabrexex2g 5689* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((A 𝑉 x A {yφ} 𝑊) → {yx A φ} V)

Theoremopabex3d 5690* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
(φA V)    &   ((φ x A) → {yψ} V)       (φ → {⟨x, y⟩ ∣ (x A ψ)} V)

Theoremopabex3 5691* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
A V    &   (x A → {yφ} V)       {⟨x, y⟩ ∣ (x A φ)} V

Theoremiunex 5692* The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B(x). (Contributed by NM, 13-Oct-2003.)
A V    &   B V        x A B V

Theoremabrexex2 5693* Existence of an existentially restricted class abstraction. φ is normally has free-variable parameters x and y. See also abrexex 5686. (Contributed by NM, 12-Sep-2004.)
A V    &   {yφ} V       {yx A φ} V

Theoremabexssex 5694* Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in φ. (Contributed by NM, 29-Jul-2006.)
A V    &   {yφ} V       {yx(xA φ)} V

Theoremabexex 5695* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
A V    &   (φx A)    &   {yφ} V       {yxφ} V

Theoremoprabexd 5696* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
(φA V)    &   (φB V)    &   ((φ (x A y B)) → ∃*zψ)    &   (φ𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) ψ)})       (φ𝐹 V)

Theoremoprabex 5697* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
A V    &   B V    &   ((x A y B) → ∃*zφ)    &   𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)}       𝐹 V

Theoremoprabex3 5698* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
𝐻 V    &   𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}       𝐹 V

Theoremoprabrexex2 5699* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
A V    &   {⟨⟨x, y⟩, z⟩ ∣ φ} V       {⟨⟨x, y⟩, z⟩ ∣ w A φ} V

Theoremab2rexex 5700* Existence of a class abstraction of existentially restricted sets. Variables x and y are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(x, y). See comments for abrexex 5686. (Contributed by NM, 20-Sep-2011.)
A V    &   B V       {zx A y B z = 𝐶} V

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