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Theorem xpcomen 7185
Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpcomen.1  |-  A  e. 
_V
xpcomen.2  |-  B  e. 
_V
Assertion
Ref Expression
xpcomen  |-  ( A  X.  B )  ~~  ( B  X.  A
)

Proof of Theorem xpcomen
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpcomen.1 . . 3  |-  A  e. 
_V
2 xpcomen.2 . . 3  |-  B  e. 
_V
31, 2xpex 4976 . 2  |-  ( A  X.  B )  e. 
_V
42, 1xpex 4976 . 2  |-  ( B  X.  A )  e. 
_V
5 eqid 2430 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } )  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )
65xpcomf1o 7183 . 2  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)
7 f1oen2g 7110 . 2  |-  ( ( ( A  X.  B
)  e.  _V  /\  ( B  X.  A
)  e.  _V  /\  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
) )  ->  ( A  X.  B )  ~~  ( B  X.  A
) )
83, 4, 6, 7mp3an 1279 1  |-  ( A  X.  B )  ~~  ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2943   {csn 3801   U.cuni 4002   class class class wbr 4199    e. cmpt 4253    X. cxp 4862   `'ccnv 4863   -1-1-onto->wf1o 5439    ~~ cen 7092
This theorem is referenced by:  xpcomeng  7186  hashxplem  11679
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-1st 6335  df-2nd 6336  df-en 7096
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