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Theorem xpexcnv 26992
Description: A condition where the converse of xpex 4754 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 4892 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  dom  (  A  X.  B
)  e.  _V )
2 dmxp 4850 . . . 4  |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )
32eleq1d 2322 . . 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 1621    =/= wne 2419   _Vcvv 2740   (/)c0 3397    X. cxp 4624   dom cdm 4626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645
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