P. 62 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	abrexex2 6101*	Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6094. (Contributed by NM, 12-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
Theorem	abexssex 6102*	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    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V
Theorem	abexex 6103*	A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V
Theorem	oprabexd 6104*	Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓)    &   ⊢ (𝜑 → 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)})    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	oprabex 6105*	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
Theorem	oprabex3 6106*	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)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}    ⇒   ⊢ 𝐹 ∈ V
Theorem	oprabrexex2 6107*	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
Theorem	ab2rexex 6108*	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 6094. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
Theorem	ab2rexex2 6109*	Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 6101. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ {𝑧 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V
Theorem	xpexgALT 6110	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4723 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
Theorem	offval3 6111*	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 6112	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 6113*	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 6114, allowing it to be used as a function or structure argument. By ofmresval 6070, 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 6114	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)
2.6.15  First and second members of an ordered pair
Syntax	c1st 6115	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 6116	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 6117	Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6123 proves that it does this. For example, (1st ‘⟨ 3 , 4 ⟩) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5090 and op1stb 4461). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
Definition	df-2nd 6118	Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6124 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5093 and op2ndb 5092). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
Theorem	1stvalg 6119	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 6120	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 6121	The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (1st ‘∅) = ∅
Theorem	2nd0 6122	The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (2nd ‘∅) = ∅
Theorem	op1st 6123	Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Theorem	op2nd 6124	Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
Theorem	op1std 6125	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 6126	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 6127	Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
Theorem	op2ndg 6128	Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
Theorem	ot1stg 6129	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 6129, ot2ndg 6130, ot3rdgg 6131.) (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 6130	Extract the second member of an ordered triple. (See ot1stg 6129 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 6131	Extract the third member of an ordered triple. (See ot1stg 6129 comment.) (Contributed by NM, 3-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Theorem	1stval2 6132	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 6133	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 6134	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 6135	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 6136	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 6137	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 6138*	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 6139*	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 6140	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 6141	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 6142	Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)
Theorem	xp2nd 6143	Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)
Theorem	1stexg 6144	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)
Theorem	2ndexg 6145	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)
Theorem	elxp6 6146	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5096. (Contributed by NM, 9-Oct-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	elxp7 6147	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5096. (Contributed by NM, 19-Aug-2006.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
Theorem	oprssdmm 6148*	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 6149	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 6150*	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 6151	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 6152	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 6153	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 6154	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 6155	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 6156*	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 6157	Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
Theorem	1st2nd 6158	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 6159	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 6160	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 6161	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 6162*	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 6163*	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 6164	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2964 that avoids the existential quantifiers of copsexg 4227). (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 6165	Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 3058). (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 6166*	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 6167*	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 6168*	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 6169*	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 6170*	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	dfxp3 6171*	Define the cross product of three classes. Compare df-xp 4615. (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 6172*	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 6173*	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 6174*	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 6175*	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 6176*	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 6177*	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 6178*	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 6179*	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	mpofvex 6180*	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	fnmpoi 6181*	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 6182*	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	mpofvexi 6183*	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 6184*	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 6185*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6182. (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 6186*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6182. 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 6187*	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 6188*	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 6189*	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 6190*	Weak version of mpoex 6191 that holds without ax-coll 4102. 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 6191*	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 6192*	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 6193*	Composition of two functions. Variation of fmptco 5660 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 6194*	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 6195*	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 6196*	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 6197*	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 6198	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 6199	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 6200*	Alternate definition for the maps-to notation df-mpo 5856 (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⟩}