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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoprabex3 7901* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
š» āˆˆ V    &   š¹ = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ ((š‘„ āˆˆ (š» Ɨ š») āˆ§ š‘¦ āˆˆ (š» Ɨ š»)) āˆ§ āˆƒš‘¤āˆƒš‘£āˆƒš‘¢āˆƒš‘“((š‘„ = āŸØš‘¤, š‘£āŸ© āˆ§ š‘¦ = āŸØš‘¢, š‘“āŸ©) āˆ§ š‘§ = š‘…))}    ā‡’   š¹ āˆˆ V
 
Theoremoprabrexex2 7902* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
š“ āˆˆ V    &   {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} āˆˆ V    ā‡’   {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ āˆƒš‘¤ āˆˆ š“ šœ‘} āˆˆ V
 
Theoremab2rexex 7903* 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 7886. (Contributed by NM, 20-Sep-2011.)
š“ āˆˆ V    &   šµ āˆˆ V    ā‡’   {š‘§ āˆ£ āˆƒš‘„ āˆˆ š“ āˆƒš‘¦ āˆˆ šµ š‘§ = š¶} āˆˆ V
 
Theoremab2rexex2 7904* Existence of an existentially restricted class abstraction. šœ‘ normally has free-variable parameters š‘„, š‘¦, and š‘§. Compare abrexex2 7893. (Contributed by NM, 20-Sep-2011.)
š“ āˆˆ V    &   šµ āˆˆ V    &   {š‘§ āˆ£ šœ‘} āˆˆ V    ā‡’   {š‘§ āˆ£ āˆƒš‘„ āˆˆ š“ āˆƒš‘¦ āˆˆ šµ šœ‘} āˆˆ V
 
TheoremxpexgALT 7905 Alternate proof of xpexg 7675 requiring Replacement (ax-rep 5241) but not Power Set (ax-pow 5319). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (š“ Ɨ šµ) āˆˆ V)
 
Theoremoffval3 7906* General value of (š¹ āˆ˜f š‘…šŗ) with no assumptions on functionality of š¹ and šŗ. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((š¹ āˆˆ š‘‰ āˆ§ šŗ āˆˆ š‘Š) ā†’ (š¹ āˆ˜f š‘…šŗ) = (š‘„ āˆˆ (dom š¹ āˆ© dom šŗ) ā†¦ ((š¹ā€˜š‘„)š‘…(šŗā€˜š‘„))))
 
Theoremoffres 7907 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((š¹ āˆˆ š‘‰ āˆ§ šŗ āˆˆ š‘Š) ā†’ ((š¹ āˆ˜f š‘…šŗ) ā†¾ š·) = ((š¹ ā†¾ š·) āˆ˜f š‘…(šŗ ā†¾ š·)))
 
Theoremofmres 7908* Equivalent expressions for a restriction of the function operation map. Unlike āˆ˜f š‘… which is a proper class, ( āˆ˜f š‘… ā†¾ (š“ Ɨ šµ)) can be a set by ofmresex 7909, allowing it to be used as a function or structure argument. By ofmresval 7624, the restricted operation map values are the same as the original values, allowing theorems for āˆ˜f š‘… to be reused. (Contributed by NM, 20-Oct-2014.)
( āˆ˜f š‘… ā†¾ (š“ Ɨ šµ)) = (š‘“ āˆˆ š“, š‘” āˆˆ šµ ā†¦ (š‘“ āˆ˜f š‘…š‘”))
 
Theoremofmresex 7909 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(šœ‘ ā†’ š“ āˆˆ š‘‰)    &   (šœ‘ ā†’ šµ āˆˆ š‘Š)    ā‡’   (šœ‘ ā†’ ( āˆ˜f š‘… ā†¾ (š“ Ɨ šµ)) āˆˆ V)
 
2.4.8  First and second members of an ordered pair
 
Syntaxc1st 7910 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 7911 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 7912 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 7920 proves that it does this. For example, (1st ā€˜āŸØ3, 4āŸ©) = 3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 6174 and op1stb 5427). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (š‘„ āˆˆ V ā†¦ āˆŖ dom {š‘„})
 
Definitiondf-2nd 7913 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 7921 proves that it does this. For example, (2nd ā€˜āŸØ3, 4āŸ©) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6177 and op2ndb 6176). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (š‘„ āˆˆ V ā†¦ āˆŖ ran {š‘„})
 
Theorem1stval 7914 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.)
(1st ā€˜š“) = āˆŖ dom {š“}
 
Theorem2ndval 7915 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.)
(2nd ā€˜š“) = āˆŖ ran {š“}
 
Theorem1stnpr 7916 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
(Ā¬ š“ āˆˆ (V Ɨ V) ā†’ (1st ā€˜š“) = āˆ…)
 
Theorem2ndnpr 7917 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
(Ā¬ š“ āˆˆ (V Ɨ V) ā†’ (2nd ā€˜š“) = āˆ…)
 
Theorem1st0 7918 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ā€˜āˆ…) = āˆ…
 
Theorem2nd0 7919 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ā€˜āˆ…) = āˆ…
 
Theoremop1st 7920 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
š“ āˆˆ V    &   šµ āˆˆ V    ā‡’   (1st ā€˜āŸØš“, šµāŸ©) = š“
 
Theoremop2nd 7921 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
š“ āˆˆ V    &   šµ āˆˆ V    ā‡’   (2nd ā€˜āŸØš“, šµāŸ©) = šµ
 
Theoremop1std 7922 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
š“ āˆˆ V    &   šµ āˆˆ V    ā‡’   (š¶ = āŸØš“, šµāŸ© ā†’ (1st ā€˜š¶) = š“)
 
Theoremop2ndd 7923 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
š“ āˆˆ V    &   šµ āˆˆ V    ā‡’   (š¶ = āŸØš“, šµāŸ© ā†’ (2nd ā€˜š¶) = šµ)
 
Theoremop1stg 7924 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (1st ā€˜āŸØš“, šµāŸ©) = š“)
 
Theoremop2ndg 7925 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (2nd ā€˜āŸØš“, šµāŸ©) = šµ)
 
Theoremot1stg 7926 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 7926, ot2ndg 7927, ot3rdg 7928.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š āˆ§ š¶ āˆˆ š‘‹) ā†’ (1st ā€˜(1st ā€˜āŸØš“, šµ, š¶āŸ©)) = š“)
 
Theoremot2ndg 7927 Extract the second member of an ordered triple. (See ot1stg 7926 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š āˆ§ š¶ āˆˆ š‘‹) ā†’ (2nd ā€˜(1st ā€˜āŸØš“, šµ, š¶āŸ©)) = šµ)
 
Theoremot3rdg 7928 Extract the third member of an ordered triple. (See ot1stg 7926 comment.) (Contributed by NM, 3-Apr-2015.)
(š¶ āˆˆ š‘‰ ā†’ (2nd ā€˜āŸØš“, šµ, š¶āŸ©) = š¶)
 
Theorem1stval2 7929 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“ āˆˆ (V Ɨ V) ā†’ (1st ā€˜š“) = āˆ© āˆ© š“)
 
Theorem2ndval2 7930 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“ āˆˆ (V Ɨ V) ā†’ (2nd ā€˜š“) = āˆ© āˆ© āˆ© ā—”{š“})
 
Theoremoteqimp 7931 The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(š‘‡ = āŸØš“, šµ, š¶āŸ© ā†’ ((š“ āˆˆ š‘‹ āˆ§ šµ āˆˆ š‘Œ āˆ§ š¶ āˆˆ š‘) ā†’ ((1st ā€˜(1st ā€˜š‘‡)) = š“ āˆ§ (2nd ā€˜(1st ā€˜š‘‡)) = šµ āˆ§ (2nd ā€˜š‘‡) = š¶)))
 
Theoremfo1st 7932 The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
1st :Vā€“ontoā†’V
 
Theoremfo2nd 7933 The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
2nd :Vā€“ontoā†’V
 
Theorembr1steqg 7934 Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on š¶. (Revised by Peter Mazsa, 17-Jan-2022.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©1st š¶ ā†” š¶ = š“))
 
Theorembr2ndeqg 7935 Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on š¶. (Revised by Peter Mazsa, 17-Jan-2022.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©2nd š¶ ā†” š¶ = šµ))
 
Theoremf1stres 7936 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 ā†¾ (š“ Ɨ šµ)):(š“ Ɨ šµ)āŸ¶š“
 
Theoremf2ndres 7937 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 ā†¾ (š“ Ɨ šµ)):(š“ Ɨ šµ)āŸ¶šµ
 
Theoremfo1stres 7938 Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
(šµ ā‰  āˆ… ā†’ (1st ā†¾ (š“ Ɨ šµ)):(š“ Ɨ šµ)ā€“ontoā†’š“)
 
Theoremfo2ndres 7939 Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
(š“ ā‰  āˆ… ā†’ (2nd ā†¾ (š“ Ɨ šµ)):(š“ Ɨ šµ)ā€“ontoā†’šµ)
 
Theorem1st2val 7940* Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
({āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ š‘§ = š‘„}ā€˜š“) = (1st ā€˜š“)
 
Theorem2nd2val 7941* Value of an alternate definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
({āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ š‘§ = š‘¦}ā€˜š“) = (2nd ā€˜š“)
 
Theorem1stcof 7942 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
(š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ (1st āˆ˜ š¹):š“āŸ¶šµ)
 
Theorem2ndcof 7943 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
(š¹:š“āŸ¶(šµ Ɨ š¶) ā†’ (2nd āˆ˜ š¹):š“āŸ¶š¶)
 
Theoremxp1st 7944 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(š“ āˆˆ (šµ Ɨ š¶) ā†’ (1st ā€˜š“) āˆˆ šµ)
 
Theoremxp2nd 7945 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
(š“ āˆˆ (šµ Ɨ š¶) ā†’ (2nd ā€˜š“) āˆˆ š¶)
 
Theoremelxp6 7946 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7850. (Contributed by NM, 9-Oct-2004.)
(š“ āˆˆ (šµ Ɨ š¶) ā†” (š“ = āŸØ(1st ā€˜š“), (2nd ā€˜š“)āŸ© āˆ§ ((1st ā€˜š“) āˆˆ šµ āˆ§ (2nd ā€˜š“) āˆˆ š¶)))
 
Theoremelxp7 7947 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7850. (Contributed by NM, 19-Aug-2006.)
(š“ āˆˆ (šµ Ɨ š¶) ā†” (š“ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š“) āˆˆ šµ āˆ§ (2nd ā€˜š“) āˆˆ š¶)))
 
Theoremeqopi 7948 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
((š“ āˆˆ (š‘‰ Ɨ š‘Š) āˆ§ ((1st ā€˜š“) = šµ āˆ§ (2nd ā€˜š“) = š¶)) ā†’ š“ = āŸØšµ, š¶āŸ©)
 
Theoremxp2 7949* Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
(š“ Ɨ šµ) = {š‘„ āˆˆ (V Ɨ V) āˆ£ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ šµ)}
 
Theoremunielxp 7950 The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.)
(š“ āˆˆ (šµ Ɨ š¶) ā†’ āˆŖ š“ āˆˆ āˆŖ (šµ Ɨ š¶))
 
Theorem1st2nd2 7951 Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
(š“ āˆˆ (šµ Ɨ š¶) ā†’ š“ = āŸØ(1st ā€˜š“), (2nd ā€˜š“)āŸ©)
 
Theorem1st2ndb 7952 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
(š“ āˆˆ (V Ɨ V) ā†” š“ = āŸØ(1st ā€˜š“), (2nd ā€˜š“)āŸ©)
 
Theoremxpopth 7953 An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.)
((š“ āˆˆ (š¶ Ɨ š·) āˆ§ šµ āˆˆ (š‘… Ɨ š‘†)) ā†’ (((1st ā€˜š“) = (1st ā€˜šµ) āˆ§ (2nd ā€˜š“) = (2nd ā€˜šµ)) ā†” š“ = šµ))
 
Theoremeqop 7954 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
(š“ āˆˆ (š‘‰ Ɨ š‘Š) ā†’ (š“ = āŸØšµ, š¶āŸ© ā†” ((1st ā€˜š“) = šµ āˆ§ (2nd ā€˜š“) = š¶)))
 
Theoremeqop2 7955 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
šµ āˆˆ V    &   š¶ āˆˆ V    ā‡’   (š“ = āŸØšµ, š¶āŸ© ā†” (š“ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š“) = šµ āˆ§ (2nd ā€˜š“) = š¶)))
 
Theoremop1steq 7956* 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“ āˆˆ (š‘‰ Ɨ š‘Š) ā†’ ((1st ā€˜š“) = šµ ā†” āˆƒš‘„ š“ = āŸØšµ, š‘„āŸ©))
 
Theoremopreuopreu 7957* There is a unique ordered pair fulfilling a wff iff its components fulfil a corresponding wff. (Contributed by AV, 2-Jul-2023.)
((š‘Ž = (1st ā€˜š‘) āˆ§ š‘ = (2nd ā€˜š‘)) ā†’ (šœ“ ā†” šœ‘))    ā‡’   (āˆƒ!š‘ āˆˆ (š“ Ɨ šµ)šœ‘ ā†” āˆƒ!š‘ āˆˆ (š“ Ɨ šµ)āˆƒš‘Žāˆƒš‘(š‘ = āŸØš‘Ž, š‘āŸ© āˆ§ šœ“))
 
Theoremel2xptp 7958* A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(š“ āˆˆ ((šµ Ɨ š¶) Ɨ š·) ā†” āˆƒš‘„ āˆˆ šµ āˆƒš‘¦ āˆˆ š¶ āˆƒš‘§ āˆˆ š· š“ = āŸØš‘„, š‘¦, š‘§āŸ©)
 
Theoremel2xptp0 7959 A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((š‘‹ āˆˆ š‘ˆ āˆ§ š‘Œ āˆˆ š‘‰ āˆ§ š‘ āˆˆ š‘Š) ā†’ ((š“ āˆˆ ((š‘ˆ Ɨ š‘‰) Ɨ š‘Š) āˆ§ ((1st ā€˜(1st ā€˜š“)) = š‘‹ āˆ§ (2nd ā€˜(1st ā€˜š“)) = š‘Œ āˆ§ (2nd ā€˜š“) = š‘)) ā†” š“ = āŸØš‘‹, š‘Œ, š‘āŸ©))
 
Theorem2nd1st 7960 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
(š“ āˆˆ (šµ Ɨ š¶) ā†’ āˆŖ ā—”{š“} = āŸØ(2nd ā€˜š“), (1st ā€˜š“)āŸ©)
 
Theorem1st2nd 7961 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
((Rel šµ āˆ§ š“ āˆˆ šµ) ā†’ š“ = āŸØ(1st ā€˜š“), (2nd ā€˜š“)āŸ©)
 
Theorem1stdm 7962 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‘… āˆ§ š“ āˆˆ š‘…) ā†’ (1st ā€˜š“) āˆˆ dom š‘…)
 
Theorem2ndrn 7963 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‘… āˆ§ š“ āˆˆ š‘…) ā†’ (2nd ā€˜š“) āˆˆ ran š‘…)
 
Theorem1st2ndbr 7964 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
((Rel šµ āˆ§ š“ āˆˆ šµ) ā†’ (1st ā€˜š“)šµ(2nd ā€˜š“))
 
Theoremreleldm2 7965* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel š“ ā†’ (šµ āˆˆ dom š“ ā†” āˆƒš‘„ āˆˆ š“ (1st ā€˜š‘„) = šµ))
 
Theoremreldm 7966* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
(Rel š“ ā†’ dom š“ = ran (š‘„ āˆˆ š“ ā†¦ (1st ā€˜š‘„)))
 
Theoremreleldmdifi 7967* One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)
((Rel š“ āˆ§ šµ āŠ† š“) ā†’ (š¶ āˆˆ (dom š“ āˆ– dom šµ) ā†’ āˆƒš‘„ āˆˆ (š“ āˆ– šµ)(1st ā€˜š‘„) = š¶))
 
Theoremfunfv1st2nd 7968 The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.)
((Fun š¹ āˆ§ š‘‹ āˆˆ š¹) ā†’ (š¹ā€˜(1st ā€˜š‘‹)) = (2nd ā€˜š‘‹))
 
Theoremfunelss 7969 If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.)
((Fun š“ āˆ§ šµ āŠ† š“ āˆ§ š‘‹ āˆˆ š“) ā†’ ((1st ā€˜š‘‹) āˆˆ dom šµ ā†’ š‘‹ āˆˆ šµ))
 
Theoremfuneldmdif 7970* Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)
((Fun š“ āˆ§ šµ āŠ† š“) ā†’ (š¶ āˆˆ (dom š“ āˆ– dom šµ) ā†” āˆƒš‘„ āˆˆ (š“ āˆ– šµ)(1st ā€˜š‘„) = š¶))
 
Theoremsbcopeq1a 7971 Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3749 that avoids the existential quantifiers of copsexg 5446). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(š“ = āŸØš‘„, š‘¦āŸ© ā†’ ([(1st ā€˜š“) / š‘„][(2nd ā€˜š“) / š‘¦]šœ‘ ā†” šœ‘))
 
Theoremcsbopeq1a 7972 Equality theorem for substitution of a class š“ for an ordered pair āŸØš‘„, š‘¦āŸ© in šµ (analogue of csbeq1a 3868). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(š“ = āŸØš‘„, š‘¦āŸ© ā†’ ā¦‹(1st ā€˜š“) / š‘„ā¦Œā¦‹(2nd ā€˜š“) / š‘¦ā¦Œšµ = šµ)
 
Theoremdfopab2 7973* 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.)
{āŸØš‘„, š‘¦āŸ© āˆ£ šœ‘} = {š‘§ āˆˆ (V Ɨ V) āˆ£ [(1st ā€˜š‘§) / š‘„][(2nd ā€˜š‘§) / š‘¦]šœ‘}
 
Theoremdfoprab3s 7974* 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.)
{āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} = {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (V Ɨ V) āˆ§ [(1st ā€˜š‘¤) / š‘„][(2nd ā€˜š‘¤) / š‘¦]šœ‘)}
 
Theoremdfoprab3 7975* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
(š‘¤ = āŸØš‘„, š‘¦āŸ© ā†’ (šœ‘ ā†” šœ“))    ā‡’   {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (V Ɨ V) āˆ§ šœ‘)} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ“}
 
Theoremdfoprab4 7976* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(š‘¤ = āŸØš‘„, š‘¦āŸ© ā†’ (šœ‘ ā†” šœ“))    ā‡’   {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (š“ Ɨ šµ) āˆ§ šœ‘)} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) āˆ§ šœ“)}
 
Theoremdfoprab4f 7977* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 31-Aug-2015.)
ā„²š‘„šœ‘    &   ā„²š‘¦šœ‘    &   (š‘¤ = āŸØš‘„, š‘¦āŸ© ā†’ (šœ‘ ā†” šœ“))    ā‡’   {āŸØš‘¤, š‘§āŸ© āˆ£ (š‘¤ āˆˆ (š“ Ɨ šµ) āˆ§ šœ‘)} = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) āˆ§ šœ“)}
 
Theoremopabex2 7978* Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
(šœ‘ ā†’ š“ āˆˆ š‘‰)    &   (šœ‘ ā†’ šµ āˆˆ š‘Š)    &   ((šœ‘ āˆ§ šœ“) ā†’ š‘„ āˆˆ š“)    &   ((šœ‘ āˆ§ šœ“) ā†’ š‘¦ āˆˆ šµ)    ā‡’   (šœ‘ ā†’ {āŸØš‘„, š‘¦āŸ© āˆ£ šœ“} āˆˆ V)
 
Theoremopabn1stprc 7979* 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.)
(āˆƒš‘¦šœ‘ ā†’ {āŸØš‘„, š‘¦āŸ© āˆ£ šœ‘} āˆ‰ V)
 
Theoremopiota 7980* The property of a uniquely specified ordered pair. The proof uses properties of the ā„© description binder. (Contributed by Mario Carneiro, 21-May-2015.)
š¼ = (ā„©š‘§āˆƒš‘„ āˆˆ š“ āˆƒš‘¦ āˆˆ šµ (š‘§ = āŸØš‘„, š‘¦āŸ© āˆ§ šœ‘))    &   š‘‹ = (1st ā€˜š¼)    &   š‘Œ = (2nd ā€˜š¼)    &   (š‘„ = š¶ ā†’ (šœ‘ ā†” šœ“))    &   (š‘¦ = š· ā†’ (šœ“ ā†” šœ’))    ā‡’   (āˆƒ!š‘§āˆƒš‘„ āˆˆ š“ āˆƒš‘¦ āˆˆ šµ (š‘§ = āŸØš‘„, š‘¦āŸ© āˆ§ šœ‘) ā†’ ((š¶ āˆˆ š“ āˆ§ š· āˆˆ šµ āˆ§ šœ’) ā†” (š¶ = š‘‹ āˆ§ š· = š‘Œ)))
 
Theoremcnvoprab 7981* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.)
(š‘Ž = āŸØš‘„, š‘¦āŸ© ā†’ (šœ“ ā†” šœ‘))    &   (šœ“ ā†’ š‘Ž āˆˆ (V Ɨ V))    ā‡’   ā—”{āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} = {āŸØš‘§, š‘ŽāŸ© āˆ£ šœ“}
 
Theoremdfxp3 7982* Define the Cartesian product of three classes. Compare df-xp 5637. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
((š“ Ɨ šµ) Ɨ š¶) = {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ (š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ āˆ§ š‘§ āˆˆ š¶)}
 
Theoremelopabi 7983* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
(š‘„ = (1st ā€˜š“) ā†’ (šœ‘ ā†” šœ“))    &   (š‘¦ = (2nd ā€˜š“) ā†’ (šœ“ ā†” šœ’))    ā‡’   (š“ āˆˆ {āŸØš‘„, š‘¦āŸ© āˆ£ šœ‘} ā†’ šœ’)
 
Theoremeloprabi 7984* 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.)
(š‘„ = (1st ā€˜(1st ā€˜š“)) ā†’ (šœ‘ ā†” šœ“))    &   (š‘¦ = (2nd ā€˜(1st ā€˜š“)) ā†’ (šœ“ ā†” šœ’))    &   (š‘§ = (2nd ā€˜š“) ā†’ (šœ’ ā†” šœƒ))    ā‡’   (š“ āˆˆ {āŸØāŸØš‘„, š‘¦āŸ©, š‘§āŸ© āˆ£ šœ‘} ā†’ šœƒ)
 
Theoremmpomptsx 7985* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
(š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶) = (š‘§ āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ) ā†¦ ā¦‹(1st ā€˜š‘§) / š‘„ā¦Œā¦‹(2nd ā€˜š‘§) / š‘¦ā¦Œš¶)
 
Theoremmpompts 7986* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
(š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶) = (š‘§ āˆˆ (š“ Ɨ šµ) ā†¦ ā¦‹(1st ā€˜š‘§) / š‘„ā¦Œā¦‹(2nd ā€˜š‘§) / š‘¦ā¦Œš¶)
 
Theoremdmmpossx 7987* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   dom š¹ āŠ† āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
 
Theoremfmpox 7988* Functionality, domain and codomain of a class given by the maps-to notation, where šµ(š‘„) is not constant but depends on š‘„. (Contributed by NM, 29-Dec-2014.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š· ā†” š¹:āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)āŸ¶š·)
 
Theoremfmpo 7989* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š· ā†” š¹:(š“ Ɨ šµ)āŸ¶š·)
 
Theoremfnmpo 7990* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š‘‰ ā†’ š¹ Fn (š“ Ɨ šµ))
 
Theoremfnmpoi 7991* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    &   š¶ āˆˆ V    ā‡’   š¹ Fn (š“ Ɨ šµ)
 
Theoremdmmpo 7992* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    &   š¶ āˆˆ V    ā‡’   dom š¹ = (š“ Ɨ šµ)
 
Theoremovmpoelrn 7993* An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
š‘‚ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   ((āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š‘€ āˆ§ š‘‹ āˆˆ š“ āˆ§ š‘Œ āˆˆ šµ) ā†’ (š‘‹š‘‚š‘Œ) āˆˆ š‘€)
 
Theoremdmmpoga 7994* Domain of an operation given by the maps-to notation, closed form of dmmpo 7992. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (Proof shortened by Lammen, 29-May-2024.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š‘‰ ā†’ dom š¹ = (š“ Ɨ šµ))
 
TheoremdmmpogaOLD 7995* Obsolete version of dmmpoga 7994 as of 29-May-2024. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š‘‰ ā†’ dom š¹ = (š“ Ɨ šµ))
 
Theoremdmmpog 7996* Domain of an operation given by the maps-to notation, closed form of dmmpo 7992. 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.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   (š¶ āˆˆ š‘‰ ā†’ dom š¹ = (š“ Ɨ šµ))
 
Theoremmpoexxg 7997* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   ((š“ āˆˆ š‘… āˆ§ āˆ€š‘„ āˆˆ š“ šµ āˆˆ š‘†) ā†’ š¹ āˆˆ V)
 
Theoremmpoexg 7998* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
š¹ = (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶)    ā‡’   ((š“ āˆˆ š‘… āˆ§ šµ āˆˆ š‘†) ā†’ š¹ āˆˆ V)
 
Theoremmpoexga 7999* 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.)
((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶) āˆˆ V)
 
Theoremmpoexw 8000* Weak version of mpoex 8001 that holds without ax-rep 5241. 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.)
š“ āˆˆ V    &   šµ āˆˆ V    &   š· āˆˆ V    &   āˆ€š‘„ āˆˆ š“ āˆ€š‘¦ āˆˆ šµ š¶ āˆˆ š·    ā‡’   (š‘„ āˆˆ š“, š‘¦ āˆˆ šµ ā†¦ š¶) āˆˆ V
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46948
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