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Theorem dmmpt2g 27653
Description: Domain of a class given by the "maps to" notation, closed form of dmmpt2 6362. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
Hypothesis
Ref Expression
dmmpt2g.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpt2g  |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, V, y
Allowed substitution hints:    F( x, y)

Proof of Theorem dmmpt2g
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( C  e.  V  /\  ( x  e.  A  /\  y  e.  B
) )  ->  C  e.  V )
21ralrimivva 2743 . 2  |-  ( C  e.  V  ->  A. x  e.  A  A. y  e.  B  C  e.  V )
3 dmmpt2g.1 . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43fnmpt2 6360 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
5 fndm 5486 . 2  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
62, 4, 53syl 19 1  |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651    X. cxp 4818   dom cdm 4820    Fn wfn 5391    e. cmpt2 6024
This theorem is referenced by:  aovmpt4g  27736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291
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