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Theorem dmmpt2 6194
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 6193 . 2  |-  F  Fn  ( A  X.  B
)
4 fndm 5343 . 2  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
53, 4ax-mp 8 1  |-  dom  F  =  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    X. cxp 4687   dom cdm 4689    Fn wfn 5250    e. cmpt2 5860
This theorem is referenced by:  1div0  9425  swrd00  11451  imasvscafn  13439  imasvscaval  13440  iscnp2  16969  xkococnlem  17353  hmeofval  17449  tngtopn  18166  nghmfval  18231  1div0apr  20841  elunirnmbfm  23558  matrcl  27466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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