MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpexcnv Structured version   Unicode version

Theorem xpexcnv 6727
Description: A condition where the converse of xpex 6589 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 6716 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
2 dmxp 5211 . . . 4  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
32eleq1d 2512 . . 3  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 219 . 2  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
54imp 429 1  |-  ( ( B  =/=  (/)  /\  ( A  X.  B )  e. 
_V )  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770    X. cxp 4987   dom cdm 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000
This theorem is referenced by:  fczsupp0  6931
  Copyright terms: Public domain W3C validator