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Theorem dmmpt2 6413
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 6412 . 2  |-  F  Fn  ( A  X.  B
)
4 fndm 5536 . 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 1652    e. wcel 1725   _Vcvv 2948    X. cxp 4868   dom cdm 4870    Fn wfn 5441    e. cmpt2 6075
This theorem is referenced by:  1div0  9671  swrd00  11757  imasvscafn  13754  imasvscaval  13755  iscnp2  17295  xkococnlem  17683  ucnima  18303  ucnprima  18304  tngtopn  18683  1div0apr  21754  elunirnmbfm  24595  cshnnn0  28202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342
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