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Theorem xpcomen 6623
Description: Commutative law for equinumerosity of Cartesian 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 4582 . 2  |-  ( A  X.  B )  e. 
_V
42, 1xpex 4582 . 2  |-  ( B  X.  A )  e. 
_V
5 eqid 2095 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } )  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )
65xpcomf1o 6621 . 2  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)
7 f1oen2g 6552 . 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 1280 1  |-  ( A  X.  B )  ~~  ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1445   _Vcvv 2633   {csn 3466   U.cuni 3675   class class class wbr 3867    |-> cmpt 3921    X. cxp 4465   `'ccnv 4466   -1-1-onto->wf1o 5048    ~~ cen 6535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-en 6538
This theorem is referenced by:  xpcomeng  6624
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