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Theorem dmmpt2 6855
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 6854 . 2  |-  F  Fn  ( A  X.  B
)
4 fndm 5680 . 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 1379    e. wcel 1767   _Vcvv 3113    X. cxp 4997   dom cdm 4999    Fn wfn 5583    |-> cmpt2 6287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786
This theorem is referenced by:  1div0  10209  swrd00  12611  swrd0  12624  repsundef  12709  cshnz  12729  imasvscafn  14795  imasvscaval  14796  iscnp2  19546  xkococnlem  19987  ucnima  20611  ucnprima  20612  tngtopn  20991  1div0apr  24949  elunirnmbfm  27975
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