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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaofrss 6101* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))       (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
 
Theoremcaoftrn 6102* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))       (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
 
2.6.14  Functions (continued)
 
TheoremresfunexgALT 6103 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 5733 but requires ax-pow 4171 and ax-un 4430. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunexg 6104 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunex2g 6105 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)
 
TheoremfnexALT 6106 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 5296. This version of fnex 5734 uses ax-pow 4171 and ax-un 4430, whereas fnex 5734 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunexw 6107 Weak version of funex 5735 that holds without ax-coll 4115. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
 
Theoremmptexw 6108* Weak version of mptex 5738 that holds without ax-coll 4115. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝐴 ∈ V    &   𝐶 ∈ V    &   𝑥𝐴 𝐵𝐶       (𝑥𝐴𝐵) ∈ V
 
Theoremfunrnex 6109 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 5735. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
 
Theoremfocdmex 6110 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
 
Theoremf1dmex 6111 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:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremabrexex 6112* 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 5737, funex 5735, fnex 5734, resfunexg 5733, and funimaexg 5296. See also abrexex2 6119. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
 
Theoremabrexexg 6113* 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.)
(𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremiunexg 6114* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
 
Theoremabrexex2g 6115* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
 
Theoremopabex3d 6116* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
(𝜑𝐴 ∈ V)    &   ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
 
Theoremopabex3 6117* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremiunex 6118* 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        𝑥𝐴 𝐵 ∈ V
 
Theoremabrexex2 6119* Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6112. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 
Theoremabexssex 6120* 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       {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
 
Theoremabexex 6121* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
𝐴 ∈ V    &   (𝜑𝑥𝐴)    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝜑} ∈ V
 
Theoremoprabexd 6122* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → ∃*𝑧𝜓)    &   (𝜑𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})       (𝜑𝐹 ∈ V)
 
Theoremoprabex 6123* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}       𝐹 ∈ V
 
Theoremoprabex3 6124* 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 6125* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
𝐴 ∈ V    &   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
 
Theoremab2rexex 6126* 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 6112. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
 
Theoremab2rexex2 6127* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 6119. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 
TheoremxpexgALT 6128 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4737 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.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theoremoffval3 6129* General value of (𝐹𝑓 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
 
Theoremoffres 6130 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘𝑓 𝑅(𝐺𝐷)))
 
Theoremofmres 6131* 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 6132, allowing it to be used as a function or structure argument. By ofmresval 6088, 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)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓𝑓 𝑅𝑔))
 
Theoremofmresex 6132 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
 
2.6.15  First and second members of an ordered pair
 
Syntaxc1st 6133 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 6134 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 6135 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6141 proves that it does this. For example, (1st ‘⟨ 3 , 4 ) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5106 and op1stb 4475). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})
 
Definitiondf-2nd 6136 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6142 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5109 and op2ndb 5108). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})
 
Theorem1stvalg 6137 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 → (1st𝐴) = dom {𝐴})
 
Theorem2ndvalg 6138 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 → (2nd𝐴) = ran {𝐴})
 
Theorem1st0 6139 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅
 
Theorem2nd0 6140 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅
 
Theoremop1st 6141 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
 
Theoremop2nd 6142 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 
Theoremop1std 6143 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)
 
Theoremop2ndd 6144 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = 𝐵)
 
Theoremop1stg 6145 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
 
Theoremop2ndg 6146 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
 
Theoremot1stg 6147 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 6147, ot2ndg 6148, ot3rdgg 6149.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
 
Theoremot2ndg 6148 Extract the second member of an ordered triple. (See ot1stg 6147 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
 
Theoremot3rdgg 6149 Extract the third member of an ordered triple. (See ot1stg 6147 comment.) (Contributed by NM, 3-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
 
Theorem1stval2 6150 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) → (1st𝐴) = 𝐴)
 
Theorem2ndval2 6151 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) → (2nd𝐴) = {𝐴})
 
Theoremfo1st 6152 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
 
Theoremfo2nd 6153 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
 
Theoremf1stres 6154 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
 
Theoremf2ndres 6155 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
 
Theoremfo1stresm 6156* Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
 
Theoremfo2ndresm 6157* Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
 
Theorem1stcof 6158 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (1st𝐹):𝐴𝐵)
 
Theorem2ndcof 6159 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd𝐹):𝐴𝐶)
 
Theoremxp1st 6160 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
 
Theoremxp2nd 6161 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 ∈ (𝐵 × 𝐶) → (2nd𝐴) ∈ 𝐶)
 
Theorem1stexg 6162 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (1st𝐴) ∈ V)
 
Theorem2ndexg 6163 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(𝐴𝑉 → (2nd𝐴) ∈ V)
 
Theoremelxp6 6164 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5112. (Contributed by NM, 9-Oct-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremelxp7 6165 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5112. (Contributed by NM, 19-Aug-2006.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 
Theoremoprssdmm 6166* Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
((𝜑𝑢𝑆) → ∃𝑣 𝑣𝑢)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   (𝜑 → Rel 𝐹)       (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
 
Theoremeqopi 6167 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
 
Theoremxp2 6168* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}
 
Theoremunielxp 6169 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (𝐵 × 𝐶) → 𝐴 (𝐵 × 𝐶))
 
Theorem1st2nd2 6170 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 = ⟨(1st𝐴), (2nd𝐴)⟩)
 
Theoremxpopth 6171 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
 
Theoremeqop 6172 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⟩ ↔ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))
 
Theoremeqop2 6173 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) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)))
 
Theoremop1steq 6174* 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) → ((1st𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩))
 
Theorem2nd1st 6175 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = ⟨(2nd𝐴), (1st𝐴)⟩)
 
Theorem1st2nd 6176 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
 
Theorem1stdm 6177 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 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
 
Theorem2ndrn 6178 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 𝑅𝐴𝑅) → (2nd𝐴) ∈ ran 𝑅)
 
Theorem1st2ndbr 6179 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))
 
Theoremreleldm2 6180* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐵))
 
Theoremreldm 6181* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel 𝐴 → dom 𝐴 = ran (𝑥𝐴 ↦ (1st𝑥)))
 
Theoremsbcopeq1a 6182 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2972 that avoids the existential quantifiers of copsexg 4241). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))
 
Theoremcsbopeq1a 6183 Equality theorem for substitution of a class 𝐴 for an ordered pair 𝑥, 𝑦 in 𝐵 (analog of csbeq1a 3066). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)
 
Theoremdfopab2 6184* 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) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}
 
Theoremdfoprab3s 6185* 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) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
 
Theoremdfoprab3 6186* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremdfoprab4 6187* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 
Theoremdfoprab4f 6188* 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 
Theoremdfxp3 6189* Define the cross product of three classes. Compare df-xp 4629. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)}
 
Theoremelopabi 6190* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
(𝑥 = (1st𝐴) → (𝜑𝜓))    &   (𝑦 = (2nd𝐴) → (𝜓𝜒))       (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
 
Theoremeloprabi 6191* 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 ‘(1st𝐴)) → (𝜑𝜓))    &   (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))    &   (𝑧 = (2nd𝐴) → (𝜒𝜃))       (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
 
Theoremmpomptsx 6192* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
 
Theoremmpompts 6193* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
(𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
 
Theoremdmmpossx 6194* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
 
Theoremfmpox 6195* 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 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
 
Theoremfmpo 6196* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹:(𝐴 × 𝐵)⟶𝐷)
 
Theoremfnmpo 6197* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑉𝐹 Fn (𝐴 × 𝐵))
 
Theoremmpofvex 6198* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((∀𝑥𝑦 𝐶𝑉𝑅𝑊𝑆𝑋) → (𝑅𝐹𝑆) ∈ V)
 
Theoremfnmpoi 6199* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       𝐹 Fn (𝐴 × 𝐵)
 
Theoremdmmpo 6200* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       dom 𝐹 = (𝐴 × 𝐵)
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