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Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreliun 4401 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
(Rel x A Bx A Rel B)

Theoremreliin 4402 An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
(x A Rel B → Rel x A B)

Theoremreluni 4403* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
(Rel Ax A Rel x)

Theoremrelint 4404* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
(x A Rel x → Rel A)

Theoremrel0 4405 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅

Theoremrelopabi 4406 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
A = {⟨x, y⟩ ∣ φ}       Rel A

Theoremrelopab 4407 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
Rel {⟨x, y⟩ ∣ φ}

Theoremreli 4408 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel I

Theoremrele 4409 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E

Theoremopabid2 4410* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel A → {⟨x, y⟩ ∣ ⟨x, y A} = A)

Theoreminopab 4411* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ψ)}

Theoremdifopab 4412* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ¬ ψ)}

Theoreminxp 4413 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((A × B) ∩ (𝐶 × 𝐷)) = ((A𝐶) × (B𝐷))

Theoremxpindi 4414 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
(A × (B𝐶)) = ((A × B) ∩ (A × 𝐶))

Theoremxpindir 4415 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
((AB) × 𝐶) = ((A × 𝐶) ∩ (B × 𝐶))

Theoremxpiindim 4416* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(y y A → (𝐶 × x A B) = x A (𝐶 × B))

Theoremxpriindim 4417* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(y y A → (𝐶 × (𝐷 x A B)) = ((𝐶 × 𝐷) ∩ x A (𝐶 × B)))

Theoremeliunxp 4418* Membership in a union of cross products. Analogue of elxp 4305 for nonconstant B(x). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 x A ({x} × B) ↔ xy(𝐶 = ⟨x, y (x A y B)))

Theoremopeliunxp2 4419* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
(x = 𝐶B = 𝐸)       (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))

Theoremraliunxp 4420* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4422, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
(x = ⟨y, z⟩ → (φψ))       (x y A ({y} × B)φy A z B ψ)

Theoremrexiunxp 4421* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4423, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
(x = ⟨y, z⟩ → (φψ))       (x y A ({y} × B)φy A z B ψ)

Theoremralxp 4422* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
(x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremrexxp 4423* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
(x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremdjussxp 4424* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
x A ({x} × B) ⊆ (A × V)

Theoremralxpf 4425* Version of ralxp 4422 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
yφ    &   zφ    &   xψ    &   (x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremrexxpf 4426* Version of rexxp 4423 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
yφ    &   zφ    &   xψ    &   (x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremiunxpf 4427* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
y𝐶    &   z𝐶    &   x𝐷    &   (x = ⟨y, z⟩ → 𝐶 = 𝐷)        x (A × B)𝐶 = y A z B 𝐷

Theoremopabbi2dv 4428* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2153. (Contributed by NM, 24-Feb-2014.)
Rel A    &   (φ → (⟨x, y Aψ))       (φA = {⟨x, y⟩ ∣ ψ})

Theoremrelop 4429* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
A V    &   B V       (Rel ⟨A, B⟩ ↔ xy(A = {x} B = {x, y}))

Theoremideqg 4430 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(B 𝑉 → (A I BA = B))

Theoremideq 4431 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
B V       (A I BA = B)

Theoremididg 4432 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A 𝑉A I A)

Theoremissetid 4433 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(A V ↔ A I A)

Theoremcoss1 4434 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(AB → (A𝐶) ⊆ (B𝐶))

Theoremcoss2 4435 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(AB → (𝐶A) ⊆ (𝐶B))

Theoremcoeq1 4436 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (A𝐶) = (B𝐶))

Theoremcoeq2 4437 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (𝐶A) = (𝐶B))

Theoremcoeq1i 4438 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (A𝐶) = (B𝐶)

Theoremcoeq2i 4439 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (𝐶A) = (𝐶B)

Theoremcoeq1d 4440 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremcoeq2d 4441 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremcoeq12i 4442 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)

Theoremcoeq12d 4443 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))

Theoremnfco 4444 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
xA    &   xB       x(AB)

Theorembrcog 4445* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))

Theoremopelco2g 4446* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (⟨A, B (𝐶𝐷) ↔ x(⟨A, x 𝐷 x, B 𝐶)))

Theorembrcogw 4447 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A(𝐶𝐷)B)

Theoremeqbrrdva 4448* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(φA ⊆ (𝐶 × 𝐷))    &   (φB ⊆ (𝐶 × 𝐷))    &   ((φ x 𝐶 y 𝐷) → (xAyxBy))       (φA = B)

Theorembrco 4449* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (A(𝐶𝐷)Bx(A𝐷x x𝐶B))

Theoremopelco 4450* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (⟨A, B (𝐶𝐷) ↔ x(A𝐷x x𝐶B))

Theoremcnvss 4451 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
(ABAB)

Theoremcnveq 4452 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(A = BA = B)

Theoremcnveqi 4453 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
A = B       A = B

Theoremcnveqd 4454 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(φA = B)       (φA = B)

Theoremelcnv 4455* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(A 𝑅xy(A = ⟨x, y y𝑅x))

Theoremelcnv2 4456* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(A 𝑅xy(A = ⟨x, yy, x 𝑅))

Theoremnfcnv 4457 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xA

Theoremopelcnvg 4458 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))

Theorembrcnvg 4459 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))

Theoremopelcnv 4460 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (⟨A, B 𝑅 ↔ ⟨B, A 𝑅)

Theorembrcnv 4461 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (A𝑅BB𝑅A)

Theoremcsbcnvg 4462 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
(A 𝑉A / x𝐹 = A / x𝐹)

Theoremcnvco 4463 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = (BA)

Theoremcnvuni 4464* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
A = x A x

Theoremdfdm3 4465* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
dom A = {xyx, y A}

Theoremdfrn2 4466* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran A = {yx xAy}

Theoremdfrn3 4467* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
ran A = {yxx, y A}

Theoremelrn2g 4468* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bxx, A B))

Theoremelrng 4469* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bx xBA))

Theoremdfdm4 4470 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom A = ran A

Theoremdfdmf 4471* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       dom A = {xy xAy}

Theoremcsbdmg 4472 Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
(A 𝑉A / xdom B = dom A / xB)

Theoremeldmg 4473* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom By ABy))

Theoremeldm2g 4474* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom ByA, y B))

Theoremeldm 4475* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
A V       (A dom By ABy)

Theoremeldm2 4476* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
A V       (A dom ByA, y B)

Theoremdmss 4477 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(AB → dom A ⊆ dom B)

Theoremdmeq 4478 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(A = B → dom A = dom B)

Theoremdmeqi 4479 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
A = B       dom A = dom B

Theoremdmeqd 4480 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → dom A = dom B)

Theoremopeldm 4481 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (⟨A, B 𝐶A dom 𝐶)

Theorembreldm 4482 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (A𝑅BA dom 𝑅)

Theoremopeldmg 4483 Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
((A 𝑉 B 𝑊) → (⟨A, B 𝐶A dom 𝐶))

Theorembreldmg 4484 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)

Theoremdmun 4485 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (AB) = (dom A ∪ dom B)

Theoremdmin 4486 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
dom (AB) ⊆ (dom A ∩ dom B)

Theoremdmiun 4487 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom x A B = x A dom B

Theoremdmuni 4488* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
dom A = x A dom x

Theoremdmopab 4489* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
dom {⟨x, y⟩ ∣ φ} = {xyφ}

Theoremdmopabss 4490* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨x, y⟩ ∣ (x A φ)} ⊆ A

Theoremdmopab3 4491* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)

Theoremdm0 4492 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom ∅ = ∅

Theoremdmi 4493 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom I = V

Theoremdmv 4494 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V

Theoremdm0rn0 4495 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
(dom A = ∅ ↔ ran A = ∅)

Theoremreldm0 4496 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel A → (A = ∅ ↔ dom A = ∅))

Theoremdmmrnm 4497* A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
(x x dom Ay y ran A)

Theoremdmxpm 4498* The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(x x B → dom (A × B) = A)

Theoremdmxpinm 4499* The domain of the intersection of two square cross products. Unlike dmin 4486, equality holds. (Contributed by NM, 29-Jan-2008.)
(x x (AB) → dom ((A × A) ∩ (B × B)) = (AB))

Theoremxpid11m 4500* The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
((x x A x x B) → ((A × A) = (B × B) ↔ A = B))

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