P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvsn 5301	The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ({⟨A, B⟩}‘A) = B
Theorem	fvsng 5302	The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ({⟨A, B⟩}‘A) = B)
Theorem	fvsnun1 5303	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5304. (Contributed by NM, 23-Sep-2007.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ 𝐺 = ({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A})))    ⇒   ⊢ (𝐺‘A) = B
Theorem	fvsnun2 5304	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5303. (Contributed by NM, 23-Sep-2007.)
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ 𝐺 = ({⟨A, B⟩} ∪ (𝐹 ↾ (𝐶 ∖ {A})))    ⇒   ⊢ (𝐷 ∈ (𝐶 ∖ {A}) → (𝐺‘𝐷) = (𝐹‘𝐷))
Theorem	fsnunf 5305	Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)
Theorem	fsnunfv 5306	Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
Theorem	fsnunres 5307	Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)
Theorem	fvpr1 5308	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
⊢ A ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (A ≠ B → ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩}‘A) = 𝐶)
Theorem	fvpr2 5309	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
⊢ B ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (A ≠ B → ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩}‘B) = 𝐷)
Theorem	fvpr1g 5310	The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
⊢ ((A ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ A ≠ B) → ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩}‘A) = 𝐶)
Theorem	fvpr2g 5311	The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
⊢ ((B ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ A ≠ B) → ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩}‘B) = 𝐷)
Theorem	fvtp1g 5312	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
⊢ (((A ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (A ≠ B ∧ A ≠ 𝐶)) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘A) = 𝐷)
Theorem	fvtp2g 5313	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
⊢ (((B ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (A ≠ B ∧ B ≠ 𝐶)) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘B) = 𝐸)
Theorem	fvtp3g 5314	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
⊢ (((𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (A ≠ 𝐶 ∧ B ≠ 𝐶)) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
Theorem	fvtp1 5315	The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
⊢ A ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ ((A ≠ B ∧ A ≠ 𝐶) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘A) = 𝐷)
Theorem	fvtp2 5316	The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
⊢ B ∈ V    &   ⊢ 𝐸 ∈ V    ⇒   ⊢ ((A ≠ B ∧ B ≠ 𝐶) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘B) = 𝐸)
Theorem	fvtp3 5317	The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ((A ≠ 𝐶 ∧ B ≠ 𝐶) → ({⟨A, 𝐷⟩, ⟨B, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
Theorem	fvconst2g 5318	The value of a constant function. (Contributed by NM, 20-Aug-2005.)
⊢ ((B ∈ 𝐷 ∧ 𝐶 ∈ A) → ((A × {B})‘𝐶) = B)
Theorem	fconst2g 5319	A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
⊢ (B ∈ 𝐶 → (𝐹:A⟶{B} ↔ 𝐹 = (A × {B})))
Theorem	fvconst2 5320	The value of a constant function. (Contributed by NM, 16-Apr-2005.)
⊢ B ∈ V    ⇒   ⊢ (𝐶 ∈ A → ((A × {B})‘𝐶) = B)
Theorem	fconst2 5321	A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
⊢ B ∈ V    ⇒   ⊢ (𝐹:A⟶{B} ↔ 𝐹 = (A × {B}))
Theorem	fconstfvm 5322*	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (Contributed by Jim Kingdon, 8-Jan-2019.)
⊢ (∃y y ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B)))
Theorem	fconst3m 5323*	Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B}))))
Theorem	fconst4m 5324*	Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A)))
Theorem	resfunexg 5325	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) ∈ V)
Theorem	fnex 5326	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 resfunexg 5325. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 Fn A ∧ A ∈ B) → 𝐹 ∈ V)
Theorem	funex 5327	If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5326. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ B) → 𝐹 ∈ V)
Theorem	opabex 5328*	Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
⊢ A ∈ V    &   ⊢ (x ∈ A → ∃*yφ)    ⇒   ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ∈ V
Theorem	mptexg 5329*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (A ∈ 𝑉 → (x ∈ A ↦ B) ∈ V)
Theorem	mptex 5330*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
⊢ A ∈ V    ⇒   ⊢ (x ∈ A ↦ B) ∈ V
Theorem	fex 5331	If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
⊢ ((𝐹:A⟶B ∧ A ∈ 𝐶) → 𝐹 ∈ V)
Theorem	eufnfv 5332*	A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ∃!f(f Fn A ∧ ∀x ∈ A (f‘x) = B)
Theorem	funfvima 5333	A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
⊢ ((Fun 𝐹 ∧ B ∈ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A)))
Theorem	funfvima2 5334	A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
⊢ ((Fun 𝐹 ∧ A ⊆ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A)))
Theorem	funfvima3 5335	A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (A ∈ dom 𝐹 → (𝐹‘A) ∈ (𝐺 “ {A})))
Theorem	fnfvima 5336	The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
⊢ ((𝐹 Fn A ∧ 𝑆 ⊆ A ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))
Theorem	rexima 5337*	Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ (x = (𝐹‘y) → (φ ↔ ψ))    ⇒   ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ))
Theorem	ralima 5338*	Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ (x = (𝐹‘y) → (φ ↔ ψ))    ⇒   ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∀x ∈ (𝐹 “ B)φ ↔ ∀y ∈ B ψ))
Theorem	idref 5339*	TODO: This is the same as issref 4650 (which has a much longer proof). Should we replace issref 4650 with this one? - NM 9-May-2016. Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)
⊢ (( I ↾ A) ⊆ 𝑅 ↔ ∀x ∈ A x𝑅x)
Theorem	elabrex 5340*	Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
⊢ B ∈ V    ⇒   ⊢ (x ∈ A → B ∈ {y ∣ ∃x ∈ A y = B})
Theorem	abrexco 5341*	Composition of two image maps 𝐶(y) and B(w). (Contributed by NM, 27-May-2013.)
⊢ B ∈ V    &   ⊢ (y = B → 𝐶 = 𝐷)    ⇒   ⊢ {x ∣ ∃y ∈ {z ∣ ∃w ∈ A z = B}x = 𝐶} = {x ∣ ∃w ∈ A x = 𝐷}
Theorem	imaiun 5342*	The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (A “ ∪ x ∈ B 𝐶) = ∪ x ∈ B (A “ 𝐶)
Theorem	imauni 5343*	The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
⊢ (A “ ∪ B) = ∪ x ∈ B (A “ x)
Theorem	fniunfv 5344*	The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
Theorem	funiunfvdm 5345*	The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5344. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))
Theorem	funiunfvdmf 5346*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5345 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ Ⅎx𝐹    ⇒   ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))
Theorem	eluniimadm 5347*	Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ (𝐹 Fn A → (B ∈ ∪ (𝐹 “ A) ↔ ∃x ∈ A B ∈ (𝐹‘x)))
Theorem	elunirn 5348*	Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
⊢ (Fun 𝐹 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
Theorem	fnunirn 5349*	Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐹 Fn 𝐼 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ 𝐼 A ∈ (𝐹‘x)))
Theorem	dff13 5350*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1veqaeq 5351	If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	dff13f 5352*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
⊢ Ⅎx𝐹    &   ⊢ Ⅎy𝐹    ⇒   ⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1mpt 5353*	Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐹 = (x ∈ A ↦ 𝐶)    &   ⊢ (x = y → 𝐶 = 𝐷)    ⇒   ⊢ (𝐹:A–1-1→B ↔ (∀x ∈ A 𝐶 ∈ B ∧ ∀x ∈ A ∀y ∈ A (𝐶 = 𝐷 → x = y)))
Theorem	f1fveq 5354	Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷))
Theorem	f1elima 5355	Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹:A–1-1→B ∧ 𝑋 ∈ A ∧ 𝑌 ⊆ A) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))
Theorem	f1imass 5356	Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
Theorem	f1imaeq 5357	Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))
Theorem	f1imapss 5358	Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷))
Theorem	dff1o6 5359*	A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
⊢ (𝐹:A–1-1-onto→B ↔ (𝐹 Fn A ∧ ran 𝐹 = B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1ocnvfv1 5360	The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv2 5361	The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv 5362	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶))
Theorem	f1ocnvfvb 5363	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A ∧ 𝐷 ∈ B) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶))
Theorem	f1ocnvdm 5364	The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (◡𝐹‘𝐶) ∈ A)
Theorem	f1ocnvfvrneq 5365	If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	fcof1 5366	An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:A⟶B ∧ (𝑅 ∘ 𝐹) = ( I ↾ A)) → 𝐹:A–1-1→B)
Theorem	fcofo 5367	An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:A⟶B ∧ 𝑆:B⟶A ∧ (𝐹 ∘ 𝑆) = ( I ↾ B)) → 𝐹:A–onto→B)
Theorem	cbvfo 5368*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹‘x) = y → (φ ↔ ψ))    ⇒   ⊢ (𝐹:A–onto→B → (∀x ∈ A φ ↔ ∀y ∈ B ψ))
Theorem	cbvexfo 5369*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
⊢ ((𝐹‘x) = y → (φ ↔ ψ))    ⇒   ⊢ (𝐹:A–onto→B → (∃x ∈ A φ ↔ ∃y ∈ B ψ))
Theorem	cocan1 5370	An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:B–1-1→𝐶 ∧ 𝐻:A⟶B ∧ 𝐾:A⟶B) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))
Theorem	cocan2 5371	A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:A–onto→B ∧ 𝐻 Fn B ∧ 𝐾 Fn B) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))
Theorem	fcof1o 5372	Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (((𝐹:A⟶B ∧ 𝐺:B⟶A) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ B) ∧ (𝐺 ∘ 𝐹) = ( I ↾ A))) → (𝐹:A–1-1-onto→B ∧ ◡𝐹 = 𝐺))
Theorem	foeqcnvco 5373	Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ ((𝐹:A–onto→B ∧ 𝐺:A–onto→B) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ B)))
Theorem	f1eqcocnv 5374	Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ ((𝐹:A–1-1→B ∧ 𝐺:A–1-1→B) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ A)))
Theorem	fliftrel 5375*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆))
Theorem	fliftel 5376*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (𝐶𝐹𝐷 ↔ ∃x ∈ 𝑋 (𝐶 = A ∧ 𝐷 = B)))
Theorem	fliftel1 5377*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ ((φ ∧ x ∈ 𝑋) → A𝐹B)
Theorem	fliftcnv 5378*	Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → ◡𝐹 = ran (x ∈ 𝑋 ↦ ⟨B, A⟩))
Theorem	fliftfun 5379*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = y → A = 𝐶)    &   ⊢ (x = y → B = 𝐷)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ ∀x ∈ 𝑋 ∀y ∈ 𝑋 (A = 𝐶 → B = 𝐷)))
Theorem	fliftfund 5380*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = y → A = 𝐶)    &   ⊢ (x = y → B = 𝐷)    &   ⊢ ((φ ∧ (x ∈ 𝑋 ∧ y ∈ 𝑋 ∧ A = 𝐶)) → B = 𝐷)    ⇒   ⊢ (φ → Fun 𝐹)
Theorem	fliftfuns 5381*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ ∀y ∈ 𝑋 ∀z ∈ 𝑋 (⦋y / x⦌A = ⦋z / x⦌A → ⦋y / x⦌B = ⦋z / x⦌B)))
Theorem	fliftf 5382*	The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ 𝐹:ran (x ∈ 𝑋 ↦ A)⟶𝑆))
Theorem	fliftval 5383*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = 𝑌 → A = 𝐶)    &   ⊢ (x = 𝑌 → B = 𝐷)    &   ⊢ (φ → Fun 𝐹)    ⇒   ⊢ ((φ ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)
Theorem	isoeq1 5384	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B)))
Theorem	isoeq2 5385	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑇, 𝑆 (A, B)))
Theorem	isoeq3 5386	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑇 (A, B)))
Theorem	isoeq4 5387	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (A = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, B)))
Theorem	isoeq5 5388	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (B = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (A, 𝐶)))
Theorem	nfiso 5389	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎx𝐻    &   ⊢ Ⅎx𝑅    &   ⊢ Ⅎx𝑆    &   ⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ Ⅎx 𝐻 Isom 𝑅, 𝑆 (A, B)
Theorem	isof1o 5390	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A–1-1-onto→B)
Theorem	isorel 5391	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	isoresbr 5392*	A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
⊢ ((𝐹 ↾ A) Isom 𝑅, 𝑆 (A, (𝐹 “ A)) → ∀x ∈ A ∀y ∈ A (x𝑅y → (𝐹‘x)𝑆(𝐹‘y)))
Theorem	isoid 5393	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ ( I ↾ A) Isom 𝑅, 𝑅 (A, A)
Theorem	isocnv 5394	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → ◡𝐻 Isom 𝑆, 𝑅 (B, A))
Theorem	isocnv2 5395	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(A, B))
Theorem	isores2 5396	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (B × B))(A, B))
Theorem	isores1 5397	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom (𝑅 ∩ (A × A)), 𝑆(A, B))
Theorem	isores3 5398	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Theorem	isotr 5399	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐺 Isom 𝑆, 𝑇 (B, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (A, 𝐶))
Theorem	isoini 5400	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})))