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Theorem xpexcnv 27573
Description: A condition where the converse of xpex 4981 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 5121 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
2 dmxp 5079 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32eleq1d 2501 . . 3  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 211 . 2  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
54imp 419 1  |-  ( ( B  =/=  (/)  /\  ( A  X.  B )  e. 
_V )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620    X. cxp 4867   dom cdm 4869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880
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