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Theorem dmmpt2 6656
Description: Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypotheses
Ref Expression
fmpt2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
fnmpt2i.2  |-  C  e. 
_V
Assertion
Ref Expression
dmmpt2  |-  dom  F  =  ( A  X.  B )
Distinct variable groups:    x, A, y    x, B, y
Allowed substitution hints:    C( x, y)    F( x, y)

Proof of Theorem dmmpt2
StepHypRef Expression
1 fmpt2.1 . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 fnmpt2i.2 . . 3  |-  C  e. 
_V
31, 2fnmpt2i 6655 . 2  |-  F  Fn  ( A  X.  B
)
4 fndm 5522 . 2  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
53, 4ax-mp 5 1  |-  dom  F  =  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2984    X. cxp 4850   dom cdm 4852    Fn wfn 5425    e. cmpt2 6105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590
This theorem is referenced by:  1div0  10007  swrd00  12326  swrd0  12339  repsundef  12421  cshnz  12441  imasvscafn  14487  imasvscaval  14488  iscnp2  18855  xkococnlem  19244  ucnima  19868  ucnprima  19869  tngtopn  20248  1div0apr  23673  elunirnmbfm  26680
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