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Theorem xpcomen 6886
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
StepHypRef Expression
1 xpcomen.1 . . 3  |-  A  e. 
_V
2 xpcomen.2 . . 3  |-  B  e. 
_V
31, 2xpex 4754 . 2  |-  ( A  X.  B )  e. 
_V
42, 1xpex 4754 . 2  |-  ( B  X.  A )  e. 
_V
5 eqid 2256 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } )  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )
65xpcomf1o 6884 . 2  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)
7 f1oen2g 6811 . 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 1282 1  |-  ( A  X.  B )  ~~  ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2740   {csn 3581   U.cuni 3768   class class class wbr 3963    e. cmpt 4017    X. cxp 4624   `'ccnv 4625   -1-1-onto->wf1o 4637    ~~ cen 6793
This theorem is referenced by:  xpcomeng  6887  hashxplem  11315
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-pow 4126  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-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-1st 6021  df-2nd 6022  df-en 6797
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