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Theorem cmpdom 25143
Description: Domain of a class given by the "maps to" notation. (Contributed by FL, 17-Feb-2008.)
Hypothesis
Ref Expression
cmpdom.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
cmpdom  |-  ( A. x  e.  A  B  e.  _V  <->  dom  F  =  A )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem cmpdom
StepHypRef Expression
1 cmpdom.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21funmpt2 5291 . 2  |-  Fun  F
3 df-fn 5258 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
41mptfng 5369 . . 3  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
5 ancom 437 . . 3  |-  ( ( dom  F  =  A  /\  Fun  F )  <-> 
( Fun  F  /\  dom  F  =  A ) )
63, 4, 53bitr4i 268 . 2  |-  ( A. x  e.  A  B  e.  _V  <->  ( dom  F  =  A  /\  Fun  F
) )
72, 6mpbiran2 885 1  |-  ( A. x  e.  A  B  e.  _V  <->  dom  F  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077   dom cdm 4689   Fun wfun 5249    Fn wfn 5250
This theorem is referenced by:  trdom2  25391  ltrdom  25401  trdom  25613
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-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-fun 5257  df-fn 5258
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