P. 64 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	offval3 6301*	General value of (𝐹 ∘𝑓 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
Theorem	offres 6302	Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘𝑓 𝑅(𝐺 ↾ 𝐷)))
Theorem	ofmres 6303*	Equivalent expressions for a restriction of the function operation map. Unlike ∘𝑓 𝑅 which is a proper class, ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 6304, allowing it to be used as a function or structure argument. By ofmresval 6252, the restricted operation map values are the same as the original values, allowing theorems for ∘𝑓 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)
⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔))
Theorem	ofmresex 6304	Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
Theorem	uchoice 6305*	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7427 (with the key difference being the change of ∃ to ∃!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
2.6.15  First and second members of an ordered pair
Syntax	c1st 6306	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 6307	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 6308	Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6314 proves that it does this. For example, (1st ‘⟨ 3 , 4 ⟩) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5220 and op1stb 4577). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
Definition	df-2nd 6309	Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6315 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5223 and op2ndb 5222). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
Theorem	1stvalg 6310	The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
Theorem	2ndvalg 6311	The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
Theorem	1st0 6312	The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (1st ‘∅) = ∅
Theorem	2nd0 6313	The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (2nd ‘∅) = ∅
Theorem	op1st 6314	Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Theorem	op2nd 6315	Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Theorem	op1std 6316	Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = 𝐴)
Theorem	op2ndd 6317	Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝐶) = 𝐵)
Theorem	op1stg 6318	Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
Theorem	op2ndg 6319	Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
Theorem	ot1stg 6320	Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6320, ot2ndg 6321, ot3rdgg 6322.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
Theorem	ot2ndg 6321	Extract the second member of an ordered triple. (See ot1stg 6320 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
Theorem	ot3rdgg 6322	Extract the third member of an ordered triple. (See ot1stg 6320 comment.) (Contributed by NM, 3-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Theorem	1stval2 6323	Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)
Theorem	2ndval2 6324	Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴})
Theorem	fo1st 6325	The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ 1st :V–onto→V
Theorem	fo2nd 6326	The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ 2nd :V–onto→V
Theorem	f1stres 6327	Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Theorem	f2ndres 6328	Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
Theorem	fo1stresm 6329*	Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
Theorem	fo2ndresm 6330*	Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
Theorem	1stcof 6331	Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵)
Theorem	2ndcof 6332	Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶)
Theorem	xp1st 6333	Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)
Theorem	xp2nd 6334	Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)
Theorem	1stexg 6335	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)
Theorem	2ndexg 6336	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)
Theorem	elxp6 6337	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5226. (Contributed by NM, 9-Oct-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	elxp7 6338	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5226. (Contributed by NM, 19-Aug-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	oprssdmm 6339*	Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → ∃𝑣 𝑣 ∈ 𝑢)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → Rel 𝐹)    ⇒   ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
Theorem	eqopi 6340	Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
Theorem	xp2 6341*	Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)}
Theorem	unielxp 6342	The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶))
Theorem	1st2nd2 6343	Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	xpopth 6344	An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵))
Theorem	eqop 6345	Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	eqop2 6346	Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
Theorem	op1steq 6347*	Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
Theorem	2nd1st 6348	Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
Theorem	1st2nd 6349	Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
Theorem	1stdm 6350	The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)
Theorem	2ndrn 6351	The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)
Theorem	1st2ndbr 6352	Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴))
Theorem	releldm2 6353*	Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵))
Theorem	reldm 6354*	An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))
Theorem	sbcopeq1a 6355	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3040 that avoids the existential quantifiers of copsexg 4338). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))
Theorem	csbopeq1a 6356	Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 3135). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)
Theorem	dfopab2 6357*	A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}
Theorem	dfoprab3s 6358*	A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}
Theorem	dfoprab3 6359*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Theorem	dfoprab4 6360*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	dfoprab4f 6361*	Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
Theorem	opabex2 6362*	Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Theorem	opabn1stprc 6363*	An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)
⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Theorem	dfxp3 6364*	Define the cross product of three classes. Compare df-xp 4733. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)}
Theorem	elopabi 6365*	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Theorem	eloprabi 6366*	A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Theorem	mpomptsx 6367*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	mpompts 6368*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
Theorem	dmmpossx 6369*	The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
Theorem	fmpox 6370*	Functionality, domain and codomain of a class given by the maps-to notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷)
Theorem	fmpo 6371*	Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷)
Theorem	fnmpo 6372*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵))
Theorem	fnmpoi 6373*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ 𝐹 Fn (𝐴 × 𝐵)
Theorem	dmmpo 6374*	Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ dom 𝐹 = (𝐴 × 𝐵)
Theorem	mpofvex 6375*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V)
Theorem	mpofvexi 6376*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (𝑅𝐹𝑆) ∈ V
Theorem	ovmpoelrn 6377*	An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀)
Theorem	dmmpoga 6378*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6374. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	dmmpog 6379*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6374. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Theorem	mpoexxg 6380*	Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexg 6381*	Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	mpoexga 6382*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
Theorem	mpoexw 6383*	Weak version of mpoex 6384 that holds without ax-coll 4205. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	mpoex 6384*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
Theorem	fnmpoovd 6385*	A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))    &   ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
Theorem	fmpoco 6386*	Composition of two functions. Variation of fmptco 5816 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))    &   ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
Theorem	oprabco 6387*	Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶))    ⇒   ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹))
Theorem	oprab2co 6388*	Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))    ⇒   ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))
Theorem	df1st2 6389*	An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Theorem	df2nd2 6390*	An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Theorem	1stconst 6391	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐵 ∈ 𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴)
Theorem	2ndconst 6392	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
⊢ (𝐴 ∈ 𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵)
Theorem	dfmpo 6393*	Alternate definition for the maps-to notation df-mpo 6028 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Theorem	cnvf1olem 6394	Lemma for cnvf1o 6395. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ◡{𝐵})) → (𝐶 ∈ ◡𝐴 ∧ 𝐵 = ∪ ◡{𝐶}))
Theorem	cnvf1o 6395*	Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)
Theorem	f2ndf 6396	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
Theorem	fo2ndf 6397	The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
Theorem	f1o2ndf1 6398	The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
Theorem	algrflem 6399	Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)
Theorem	algrflemg 6400	Lemma for algrf 12640 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵))