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Theorem xpexcnv 27058
Description: A condition where the converse of xpex 4800 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
Assertion
Ref Expression
xpexcnv  |-  ( ( B  =/=  (/)  /\  ( A  X.  B )  e. 
_V )  ->  A  e.  _V )

Proof of Theorem xpexcnv
StepHypRef Expression
1 dmexg 4938 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  dom  (  A  X.  B
)  e.  _V )
2 dmxp 4896 . . . 4  |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )
32eleq1d 2350 . . 3  |-  ( B  =/=  (/)  ->  ( dom  (  A  X.  B
)  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 212 . 2  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
54imp 420 1  |-  ( ( B  =/=  (/)  /\  ( A  X.  B )  e. 
_V )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685    =/= wne 2447   _Vcvv 2789   (/)c0 3456    X. cxp 4686   dom cdm 4688
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699
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