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Theorem cmpdom 25246
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 5307 . 2  |-  Fun  F
3 df-fn 5274 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
41mptfng 5385 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    e. cmpt 4093   dom cdm 4705   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  trdom2  25494  ltrdom  25504  trdom  25716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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