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Theorem dmmpt2 6868
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 6867 . 2  |-  F  Fn  ( A  X.  B
)
4 fndm 5684 . 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 1437    e. wcel 1867   _Vcvv 3078    X. cxp 4843   dom cdm 4845    Fn wfn 5587    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799
This theorem is referenced by:  1div0  10260  swrd00  12748  swrd0  12764  repsundef  12848  cshnz  12868  imasvscafn  15387  imasvscaval  15388  iscnp2  20179  xkococnlem  20598  ucnima  21220  ucnprima  21221  tngtopn  21582  1div0apr  25747  smatlem  28459  elunirnmbfm  28911  pfx00  38328  pfx0  38329
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