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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfof 6101 Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
ā„²š‘„š‘…    ⇒   ā„²š‘„ āˆ˜š‘“ š‘…
 
Theoremnfofr 6102 Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
ā„²š‘„š‘…    ⇒   ā„²š‘„ āˆ˜š‘Ÿ š‘…
 
Theoremoffval 6103* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š‘†    &   ((šœ‘ ∧ š‘„ ∈ š“) → (š¹ā€˜š‘„) = š¶)    &   ((šœ‘ ∧ š‘„ ∈ šµ) → (šŗā€˜š‘„) = š·)    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ) = (š‘„ ∈ š‘† ↦ (š¶š‘…š·)))
 
Theoremofrfval 6104* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š‘†    &   ((šœ‘ ∧ š‘„ ∈ š“) → (š¹ā€˜š‘„) = š¶)    &   ((šœ‘ ∧ š‘„ ∈ šµ) → (šŗā€˜š‘„) = š·)    ⇒   (šœ‘ → (š¹ āˆ˜š‘Ÿ š‘…šŗ ↔ āˆ€š‘„ ∈ š‘† š¶š‘…š·))
 
Theoremofvalg 6105 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š‘†    &   ((šœ‘ ∧ š‘‹ ∈ š“) → (š¹ā€˜š‘‹) = š¶)    &   ((šœ‘ ∧ š‘‹ ∈ šµ) → (šŗā€˜š‘‹) = š·)    &   ((šœ‘ ∧ š‘‹ ∈ š‘†) → (š¶š‘…š·) ∈ š‘ˆ)    ⇒   ((šœ‘ ∧ š‘‹ ∈ š‘†) → ((š¹ āˆ˜š‘“ š‘…šŗ)ā€˜š‘‹) = (š¶š‘…š·))
 
Theoremofrval 6106 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š‘†    &   ((šœ‘ ∧ š‘‹ ∈ š“) → (š¹ā€˜š‘‹) = š¶)    &   ((šœ‘ ∧ š‘‹ ∈ šµ) → (šŗā€˜š‘‹) = š·)    ⇒   ((šœ‘ ∧ š¹ āˆ˜š‘Ÿ š‘…šŗ ∧ š‘‹ ∈ š‘†) → š¶š‘…š·)
 
Theoremofmresval 6107 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(šœ‘ → š¹ ∈ š“)    &   (šœ‘ → šŗ ∈ šµ)    ⇒   (šœ‘ → (š¹( āˆ˜š‘“ š‘… ↾ (š“ Ɨ šµ))šŗ) = (š¹ āˆ˜š‘“ š‘…šŗ))
 
Theoremoff 6108* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘‡)) → (š‘„š‘…š‘¦) ∈ š‘ˆ)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šŗ:šµāŸ¶š‘‡)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š¶    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ):š¶āŸ¶š‘ˆ)
 
Theoremoffeq 6109* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘‡)) → (š‘„š‘…š‘¦) ∈ š‘ˆ)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šŗ:šµāŸ¶š‘‡)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š¶    &   (šœ‘ → š»:š¶āŸ¶š‘ˆ)    &   ((šœ‘ ∧ š‘„ ∈ š“) → (š¹ā€˜š‘„) = š·)    &   ((šœ‘ ∧ š‘„ ∈ šµ) → (šŗā€˜š‘„) = šø)    &   ((šœ‘ ∧ š‘„ ∈ š¶) → (š·š‘…šø) = (š»ā€˜š‘„))    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ) = š»)
 
Theoremofres 6110 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (š“ ∩ šµ) = š¶    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ) = ((š¹ ↾ š¶) āˆ˜š‘“ š‘…(šŗ ↾ š¶)))
 
Theoremoffval2 6111* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   ((šœ‘ ∧ š‘„ ∈ š“) → šµ ∈ š‘Š)    &   ((šœ‘ ∧ š‘„ ∈ š“) → š¶ ∈ š‘‹)    &   (šœ‘ → š¹ = (š‘„ ∈ š“ ↦ šµ))    &   (šœ‘ → šŗ = (š‘„ ∈ š“ ↦ š¶))    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ) = (š‘„ ∈ š“ ↦ (šµš‘…š¶)))
 
Theoremofrfval2 6112* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   ((šœ‘ ∧ š‘„ ∈ š“) → šµ ∈ š‘Š)    &   ((šœ‘ ∧ š‘„ ∈ š“) → š¶ ∈ š‘‹)    &   (šœ‘ → š¹ = (š‘„ ∈ š“ ↦ šµ))    &   (šœ‘ → šŗ = (š‘„ ∈ š“ ↦ š¶))    ⇒   (šœ‘ → (š¹ āˆ˜š‘Ÿ š‘…šŗ ↔ āˆ€š‘„ ∈ š“ šµš‘…š¶))
 
Theoremsuppssof1 6113* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(šœ‘ → (ā—”š“ ā€œ (V āˆ– {š‘Œ})) āŠ† šæ)    &   ((šœ‘ ∧ š‘£ ∈ š‘…) → (š‘Œš‘‚š‘£) = š‘)    &   (šœ‘ → š“:š·āŸ¶š‘‰)    &   (šœ‘ → šµ:š·āŸ¶š‘…)    &   (šœ‘ → š· ∈ š‘Š)    ⇒   (šœ‘ → (ā—”(š“ āˆ˜š‘“ š‘‚šµ) ā€œ (V āˆ– {š‘})) āŠ† šæ)
 
Theoremofco 6114 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
(šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn šµ)    &   (šœ‘ → š»:š·āŸ¶š¶)    &   (šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (šœ‘ → š· ∈ š‘‹)    &   (š“ ∩ šµ) = š¶    ⇒   (šœ‘ → ((š¹ āˆ˜š‘“ š‘…šŗ) ∘ š») = ((š¹ ∘ š») āˆ˜š‘“ š‘…(šŗ ∘ š»)))
 
Theoremoffveqb 6115* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹ Fn š“)    &   (šœ‘ → šŗ Fn š“)    &   (šœ‘ → š» Fn š“)    &   ((šœ‘ ∧ š‘„ ∈ š“) → (š¹ā€˜š‘„) = šµ)    &   ((šœ‘ ∧ š‘„ ∈ š“) → (šŗā€˜š‘„) = š¶)    ⇒   (šœ‘ → (š» = (š¹ āˆ˜š‘“ š‘…šŗ) ↔ āˆ€š‘„ ∈ š“ (š»ā€˜š‘„) = (šµš‘…š¶)))
 
Theoremofc12 6116 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    &   (šœ‘ → š¶ ∈ š‘‹)    ⇒   (šœ‘ → ((š“ Ɨ {šµ}) āˆ˜š‘“ š‘…(š“ Ɨ {š¶})) = (š“ Ɨ {(šµš‘…š¶)}))
 
Theoremcaofref 6117* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   ((šœ‘ ∧ š‘„ ∈ š‘†) → š‘„š‘…š‘„)    ⇒   (šœ‘ → š¹ āˆ˜š‘Ÿ š‘…š¹)
 
Theoremcaofinvl 6118* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šµ ∈ š‘Š)    &   (šœ‘ → š‘:š‘†āŸ¶š‘†)    &   (šœ‘ → šŗ = (š‘£ ∈ š“ ↦ (š‘ā€˜(š¹ā€˜š‘£))))    &   ((šœ‘ ∧ š‘„ ∈ š‘†) → ((š‘ā€˜š‘„)š‘…š‘„) = šµ)    ⇒   (šœ‘ → (šŗ āˆ˜š‘“ š‘…š¹) = (š“ Ɨ {šµ}))
 
Theoremcaofcom 6119* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šŗ:š“āŸ¶š‘†)    &   ((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘†)) → (š‘„š‘…š‘¦) = (š‘¦š‘…š‘„))    ⇒   (šœ‘ → (š¹ āˆ˜š‘“ š‘…šŗ) = (šŗ āˆ˜š‘“ š‘…š¹))
 
Theoremcaofrss 6120* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šŗ:š“āŸ¶š‘†)    &   ((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘†)) → (š‘„š‘…š‘¦ → š‘„š‘‡š‘¦))    ⇒   (šœ‘ → (š¹ āˆ˜š‘Ÿ š‘…šŗ → š¹ āˆ˜š‘Ÿ š‘‡šŗ))
 
Theoremcaoftrn 6121* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → š¹:š“āŸ¶š‘†)    &   (šœ‘ → šŗ:š“āŸ¶š‘†)    &   (šœ‘ → š»:š“āŸ¶š‘†)    &   ((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘† ∧ š‘§ ∈ š‘†)) → ((š‘„š‘…š‘¦ ∧ š‘¦š‘‡š‘§) → š‘„š‘ˆš‘§))    ⇒   (šœ‘ → ((š¹ āˆ˜š‘Ÿ š‘…šŗ ∧ šŗ āˆ˜š‘Ÿ š‘‡š») → š¹ āˆ˜š‘Ÿ š‘ˆš»))
 
2.6.14  Functions (continued)
 
TheoremresfunexgALT 6122 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 5750 but requires ax-pow 4186 and ax-un 4445. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun š“ ∧ šµ ∈ š¶) → (š“ ↾ šµ) ∈ V)
 
Theoremcofunexg 6123 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun š“ ∧ šµ ∈ š¶) → (š“ ∘ šµ) ∈ V)
 
Theoremcofunex2g 6124 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((š“ ∈ š‘‰ ∧ Fun ā—”šµ) → (š“ ∘ šµ) ∈ V)
 
TheoremfnexALT 6125 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 5312. This version of fnex 5751 uses ax-pow 4186 and ax-un 4445, whereas fnex 5751 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((š¹ Fn š“ ∧ š“ ∈ šµ) → š¹ ∈ V)
 
Theoremfunexw 6126 Weak version of funex 5752 that holds without ax-coll 4130. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
((Fun š¹ ∧ dom š¹ ∈ šµ ∧ ran š¹ ∈ š¶) → š¹ ∈ V)
 
Theoremmptexw 6127* Weak version of mptex 5755 that holds without ax-coll 4130. 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.)
š“ ∈ V    &   š¶ ∈ V    &   āˆ€š‘„ ∈ š“ šµ ∈ š¶    ⇒   (š‘„ ∈ š“ ↦ šµ) ∈ V
 
Theoremfunrnex 6128 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 5752. (Contributed by NM, 11-Nov-1995.)
(dom š¹ ∈ šµ → (Fun š¹ → ran š¹ ∈ V))
 
Theoremfocdmex 6129 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(š“ ∈ š¶ → (š¹:š“ā€“ontoā†’šµ → šµ ∈ V))
 
Theoremf1dmex 6130 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.)
((š¹:š“ā€“1-1ā†’šµ ∧ šµ ∈ š¶) → š“ ∈ V)
 
Theoremabrexex 6131* Existence of a class abstraction of existentially restricted sets. š‘„ is normally a free-variable parameter in the class expression substituted for šµ, which can be thought of as šµ(š‘„). 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 5754, funex 5752, fnex 5751, resfunexg 5750, and funimaexg 5312. See also abrexex2 6138. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
š“ ∈ V    ⇒   {š‘¦ ∣ āˆƒš‘„ ∈ š“ š‘¦ = šµ} ∈ V
 
Theoremabrexexg 6132* Existence of a class abstraction of existentially restricted sets. š‘„ is normally a free-variable parameter in šµ. The antecedent assures us that š“ is a set. (Contributed by NM, 3-Nov-2003.)
(š“ ∈ š‘‰ → {š‘¦ ∣ āˆƒš‘„ ∈ š“ š‘¦ = šµ} ∈ V)
 
Theoremiunexg 6133* The existence of an indexed union. š‘„ is normally a free-variable parameter in šµ. (Contributed by NM, 23-Mar-2006.)
((š“ ∈ š‘‰ ∧ āˆ€š‘„ ∈ š“ šµ ∈ š‘Š) → ∪ š‘„ ∈ š“ šµ ∈ V)
 
Theoremabrexex2g 6134* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((š“ ∈ š‘‰ ∧ āˆ€š‘„ ∈ š“ {š‘¦ ∣ šœ‘} ∈ š‘Š) → {š‘¦ ∣ āˆƒš‘„ ∈ š“ šœ‘} ∈ V)
 
Theoremopabex3d 6135* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
(šœ‘ → š“ ∈ V)    &   ((šœ‘ ∧ š‘„ ∈ š“) → {š‘¦ ∣ šœ“} ∈ V)    ⇒   (šœ‘ → {āŸØš‘„, š‘¦āŸ© ∣ (š‘„ ∈ š“ ∧ šœ“)} ∈ V)
 
Theoremopabex3 6136* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
š“ ∈ V    &   (š‘„ ∈ š“ → {š‘¦ ∣ šœ‘} ∈ V)    ⇒   {āŸØš‘„, š‘¦āŸ© ∣ (š‘„ ∈ š“ ∧ šœ‘)} ∈ V
 
Theoremiunex 6137* The existence of an indexed union. š‘„ is normally a free-variable parameter in the class expression substituted for šµ, which can be read informally as šµ(š‘„). (Contributed by NM, 13-Oct-2003.)
š“ ∈ V    &   šµ ∈ V    ⇒   āˆŖ š‘„ ∈ š“ šµ ∈ V
 
Theoremabrexex2 6138* Existence of an existentially restricted class abstraction. šœ‘ is normally has free-variable parameters š‘„ and š‘¦. See also abrexex 6131. (Contributed by NM, 12-Sep-2004.)
š“ ∈ V    &   {š‘¦ ∣ šœ‘} ∈ V    ⇒   {š‘¦ ∣ āˆƒš‘„ ∈ š“ šœ‘} ∈ V
 
Theoremabexssex 6139* Existence of a class abstraction with an existentially quantified expression. Both š‘„ and š‘¦ can be free in šœ‘. (Contributed by NM, 29-Jul-2006.)
š“ ∈ V    &   {š‘¦ ∣ šœ‘} ∈ V    ⇒   {š‘¦ ∣ āˆƒš‘„(š‘„ āŠ† š“ ∧ šœ‘)} ∈ V
 
Theoremabexex 6140* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
š“ ∈ V    &   (šœ‘ → š‘„ ∈ š“)    &   {š‘¦ ∣ šœ‘} ∈ V    ⇒   {š‘¦ ∣ āˆƒš‘„šœ‘} ∈ V
 
Theoremoprabexd 6141* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
(šœ‘ → š“ ∈ V)    &   (šœ‘ → šµ ∈ V)    &   ((šœ‘ ∧ (š‘„ ∈ š“ ∧ š‘¦ ∈ šµ)) → ∃*š‘§šœ“)    &   (šœ‘ → š¹ = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ ((š‘„ ∈ š“ ∧ š‘¦ ∈ šµ) ∧ šœ“)})    ⇒   (šœ‘ → š¹ ∈ V)
 
Theoremoprabex 6142* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
š“ ∈ V    &   šµ ∈ V    &   ((š‘„ ∈ š“ ∧ š‘¦ ∈ šµ) → ∃*š‘§šœ‘)    &   š¹ = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ ((š‘„ ∈ š“ ∧ š‘¦ ∈ šµ) ∧ šœ‘)}    ⇒   š¹ ∈ V
 
Theoremoprabex3 6143* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
š» ∈ V    &   š¹ = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ ((š‘„ ∈ (š» Ɨ š») ∧ š‘¦ ∈ (š» Ɨ š»)) ∧ āˆƒš‘¤āˆƒš‘£āˆƒš‘¢āˆƒš‘“((š‘„ = āŸØš‘¤, š‘£āŸ© ∧ š‘¦ = āŸØš‘¢, š‘“āŸ©) ∧ š‘§ = š‘…))}    ⇒   š¹ ∈ V
 
Theoremoprabrexex2 6144* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
š“ ∈ V    &   {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ šœ‘} ∈ V    ⇒   {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© ∣ āˆƒš‘¤ ∈ š“ šœ‘} ∈ V
 
Theoremab2rexex 6145* Existence of a class abstraction of existentially restricted sets. Variables š‘„ and š‘¦ are normally free-variable parameters in the class expression substituted for š¶, which can be thought of as š¶(š‘„, š‘¦). See comments for abrexex 6131. (Contributed by NM, 20-Sep-2011.)
š“ ∈ V    &   šµ ∈ V    ⇒   {š‘§ ∣ āˆƒš‘„ ∈ š“ āˆƒš‘¦ ∈ šµ š‘§ = š¶} ∈ V
 
Theoremab2rexex2 6146* Existence of an existentially restricted class abstraction. šœ‘ normally has free-variable parameters š‘„, š‘¦, and š‘§. Compare abrexex2 6138. (Contributed by NM, 20-Sep-2011.)
š“ ∈ V    &   šµ ∈ V    &   {š‘§ ∣ šœ‘} ∈ V    ⇒   {š‘§ ∣ āˆƒš‘„ ∈ š“ āˆƒš‘¦ ∈ šµ šœ‘} ∈ V
 
TheoremxpexgALT 6147 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4752 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.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š) → (š“ Ɨ šµ) ∈ V)
 
Theoremoffval3 6148* General value of (š¹ āˆ˜š‘“ š‘…šŗ) with no assumptions on functionality of š¹ and šŗ. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((š¹ ∈ š‘‰ ∧ šŗ ∈ š‘Š) → (š¹ āˆ˜š‘“ š‘…šŗ) = (š‘„ ∈ (dom š¹ ∩ dom šŗ) ↦ ((š¹ā€˜š‘„)š‘…(šŗā€˜š‘„))))
 
Theoremoffres 6149 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((š¹ ∈ š‘‰ ∧ šŗ ∈ š‘Š) → ((š¹ āˆ˜š‘“ š‘…šŗ) ↾ š·) = ((š¹ ↾ š·) āˆ˜š‘“ š‘…(šŗ ↾ š·)))
 
Theoremofmres 6150* Equivalent expressions for a restriction of the function operation map. Unlike āˆ˜š‘“ š‘… which is a proper class, ( āˆ˜š‘“ š‘… ↾ (š“ Ɨ šµ)) can be a set by ofmresex 6151, allowing it to be used as a function or structure argument. By ofmresval 6107, the restricted operation map values are the same as the original values, allowing theorems for āˆ˜š‘“ š‘… to be reused. (Contributed by NM, 20-Oct-2014.)
( āˆ˜š‘“ š‘… ↾ (š“ Ɨ šµ)) = (š‘“ ∈ š“, š‘” ∈ šµ ↦ (š‘“ āˆ˜š‘“ š‘…š‘”))
 
Theoremofmresex 6151 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(šœ‘ → š“ ∈ š‘‰)    &   (šœ‘ → šµ ∈ š‘Š)    ⇒   (šœ‘ → ( āˆ˜š‘“ š‘… ↾ (š“ Ɨ šµ)) ∈ V)
 
2.6.15  First and second members of an ordered pair
 
Syntaxc1st 6152 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 6153 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 6154 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6160 proves that it does this. For example, (1st ā€˜āŸØ 3 , 4 ⟩) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5122 and op1stb 4490). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (š‘„ ∈ V ↦ ∪ dom {š‘„})
 
Definitiondf-2nd 6155 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6161 proves that it does this. For example, (2nd ā€˜āŸØ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5125 and op2ndb 5124). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (š‘„ ∈ V ↦ ∪ ran {š‘„})
 
Theorem1stvalg 6156 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.)
(š“ ∈ V → (1st ā€˜š“) = ∪ dom {š“})
 
Theorem2ndvalg 6157 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.)
(š“ ∈ V → (2nd ā€˜š“) = ∪ ran {š“})
 
Theorem1st0 6158 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ā€˜āˆ…) = āˆ…
 
Theorem2nd0 6159 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ā€˜āˆ…) = āˆ…
 
Theoremop1st 6160 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
š“ ∈ V    &   šµ ∈ V    ⇒   (1st ā€˜āŸØš“, šµāŸ©) = š“
 
Theoremop2nd 6161 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
š“ ∈ V    &   šµ ∈ V    ⇒   (2nd ā€˜āŸØš“, šµāŸ©) = šµ
 
Theoremop1std 6162 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
š“ ∈ V    &   šµ ∈ V    ⇒   (š¶ = āŸØš“, šµāŸ© → (1st ā€˜š¶) = š“)
 
Theoremop2ndd 6163 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
š“ ∈ V    &   šµ ∈ V    ⇒   (š¶ = āŸØš“, šµāŸ© → (2nd ā€˜š¶) = šµ)
 
Theoremop1stg 6164 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š) → (1st ā€˜āŸØš“, šµāŸ©) = š“)
 
Theoremop2ndg 6165 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š) → (2nd ā€˜āŸØš“, šµāŸ©) = šµ)
 
Theoremot1stg 6166 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 6166, ot2ndg 6167, ot3rdgg 6168.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š ∧ š¶ ∈ š‘‹) → (1st ā€˜(1st ā€˜āŸØš“, šµ, š¶āŸ©)) = š“)
 
Theoremot2ndg 6167 Extract the second member of an ordered triple. (See ot1stg 6166 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š ∧ š¶ ∈ š‘‹) → (2nd ā€˜(1st ā€˜āŸØš“, šµ, š¶āŸ©)) = šµ)
 
Theoremot3rdgg 6168 Extract the third member of an ordered triple. (See ot1stg 6166 comment.) (Contributed by NM, 3-Apr-2015.)
((š“ ∈ š‘‰ ∧ šµ ∈ š‘Š ∧ š¶ ∈ š‘‹) → (2nd ā€˜āŸØš“, šµ, š¶āŸ©) = š¶)
 
Theorem1stval2 6169 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.)
(š“ ∈ (V Ɨ V) → (1st ā€˜š“) = ∩ ∩ š“)
 
Theorem2ndval2 6170 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.)
(š“ ∈ (V Ɨ V) → (2nd ā€˜š“) = ∩ ∩ ∩ ā—”{š“})
 
Theoremfo1st 6171 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 6172 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 6173 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 ↾ (š“ Ɨ šµ)):(š“ Ɨ šµ)āŸ¶š“
 
Theoremf2ndres 6174 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 ↾ (š“ Ɨ šµ)):(š“ Ɨ šµ)āŸ¶šµ
 
Theoremfo1stresm 6175* Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(āˆƒš‘¦ š‘¦ ∈ šµ → (1st ↾ (š“ Ɨ šµ)):(š“ Ɨ šµ)–ontoā†’š“)
 
Theoremfo2ndresm 6176* Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
(āˆƒš‘„ š‘„ ∈ š“ → (2nd ↾ (š“ Ɨ šµ)):(š“ Ɨ šµ)–ontoā†’šµ)
 
Theorem1stcof 6177 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(š¹:š“āŸ¶(šµ Ɨ š¶) → (1st ∘ š¹):š“āŸ¶šµ)
 
Theorem2ndcof 6178 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(š¹:š“āŸ¶(šµ Ɨ š¶) → (2nd ∘ š¹):š“āŸ¶š¶)
 
Theoremxp1st 6179 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(š“ ∈ (šµ Ɨ š¶) → (1st ā€˜š“) ∈ šµ)
 
Theoremxp2nd 6180 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(š“ ∈ (šµ Ɨ š¶) → (2nd ā€˜š“) ∈ š¶)
 
Theorem1stexg 6181 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(š“ ∈ š‘‰ → (1st ā€˜š“) ∈ V)
 
Theorem2ndexg 6182 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
(š“ ∈ š‘‰ → (2nd ā€˜š“) ∈ V)
 
Theoremelxp6 6183 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5128. (Contributed by NM, 9-Oct-2004.)
(š“ ∈ (šµ Ɨ š¶) ↔ (š“ = ⟨(1st ā€˜š“), (2nd ā€˜š“)⟩ ∧ ((1st ā€˜š“) ∈ šµ ∧ (2nd ā€˜š“) ∈ š¶)))
 
Theoremelxp7 6184 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5128. (Contributed by NM, 19-Aug-2006.)
(š“ ∈ (šµ Ɨ š¶) ↔ (š“ ∈ (V Ɨ V) ∧ ((1st ā€˜š“) ∈ šµ ∧ (2nd ā€˜š“) ∈ š¶)))
 
Theoremoprssdmm 6185* Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
((šœ‘ ∧ š‘¢ ∈ š‘†) → āˆƒš‘£ š‘£ ∈ š‘¢)    &   ((šœ‘ ∧ (š‘„ ∈ š‘† ∧ š‘¦ ∈ š‘†)) → (š‘„š¹š‘¦) ∈ š‘†)    &   (šœ‘ → Rel š¹)    ⇒   (šœ‘ → (š‘† Ɨ š‘†) āŠ† dom š¹)
 
Theoremeqopi 6186 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((š“ ∈ (š‘‰ Ɨ š‘Š) ∧ ((1st ā€˜š“) = šµ ∧ (2nd ā€˜š“) = š¶)) → š“ = āŸØšµ, š¶āŸ©)
 
Theoremxp2 6187* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(š“ Ɨ šµ) = {š‘„ ∈ (V Ɨ V) ∣ ((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ šµ)}
 
Theoremunielxp 6188 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(š“ ∈ (šµ Ɨ š¶) → ∪ š“ ∈ ∪ (šµ Ɨ š¶))
 
Theorem1st2nd2 6189 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
(š“ ∈ (šµ Ɨ š¶) → š“ = ⟨(1st ā€˜š“), (2nd ā€˜š“)⟩)
 
Theoremxpopth 6190 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
((š“ ∈ (š¶ Ɨ š·) ∧ šµ ∈ (š‘… Ɨ š‘†)) → (((1st ā€˜š“) = (1st ā€˜šµ) ∧ (2nd ā€˜š“) = (2nd ā€˜šµ)) ↔ š“ = šµ))
 
Theoremeqop 6191 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
(š“ ∈ (š‘‰ Ɨ š‘Š) → (š“ = āŸØšµ, š¶āŸ© ↔ ((1st ā€˜š“) = šµ ∧ (2nd ā€˜š“) = š¶)))
 
Theoremeqop2 6192 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
šµ ∈ V    &   š¶ ∈ V    ⇒   (š“ = āŸØšµ, š¶āŸ© ↔ (š“ ∈ (V Ɨ V) ∧ ((1st ā€˜š“) = šµ ∧ (2nd ā€˜š“) = š¶)))
 
Theoremop1steq 6193* 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.)
(š“ ∈ (š‘‰ Ɨ š‘Š) → ((1st ā€˜š“) = šµ ↔ āˆƒš‘„ š“ = āŸØšµ, š‘„āŸ©))
 
Theorem2nd1st 6194 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(š“ ∈ (šµ Ɨ š¶) → ∪ ā—”{š“} = ⟨(2nd ā€˜š“), (1st ā€˜š“)⟩)
 
Theorem1st2nd 6195 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
((Rel šµ ∧ š“ ∈ šµ) → š“ = ⟨(1st ā€˜š“), (2nd ā€˜š“)⟩)
 
Theorem1stdm 6196 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 š‘… ∧ š“ ∈ š‘…) → (1st ā€˜š“) ∈ dom š‘…)
 
Theorem2ndrn 6197 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 š‘… ∧ š“ ∈ š‘…) → (2nd ā€˜š“) ∈ ran š‘…)
 
Theorem1st2ndbr 6198 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
((Rel šµ ∧ š“ ∈ šµ) → (1st ā€˜š“)šµ(2nd ā€˜š“))
 
Theoremreleldm2 6199* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel š“ → (šµ ∈ dom š“ ↔ āˆƒš‘„ ∈ š“ (1st ā€˜š‘„) = šµ))
 
Theoremreldm 6200* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel š“ → dom š“ = ran (š‘„ ∈ š“ ↦ (1st ā€˜š‘„)))
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