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Theorem xpcomeng 7136
Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
Assertion
Ref Expression
xpcomeng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )

Proof of Theorem xpcomeng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 4832 . . 3  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
2 xpeq2 4833 . . 3  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
31, 2breq12d 4166 . 2  |-  ( x  =  A  ->  (
( x  X.  y
)  ~~  ( y  X.  x )  <->  ( A  X.  y )  ~~  (
y  X.  A ) ) )
4 xpeq2 4833 . . 3  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
5 xpeq1 4832 . . 3  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
64, 5breq12d 4166 . 2  |-  ( y  =  B  ->  (
( A  X.  y
)  ~~  ( y  X.  A )  <->  ( A  X.  B )  ~~  ( B  X.  A ) ) )
7 vex 2902 . . 3  |-  x  e. 
_V
8 vex 2902 . . 3  |-  y  e. 
_V
97, 8xpcomen 7135 . 2  |-  ( x  X.  y )  ~~  ( y  X.  x
)
103, 6, 9vtocl2g 2958 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153    X. cxp 4816    ~~ cen 7042
This theorem is referenced by:  xpsnen2g  7137  xpdom1g  7141  omxpen  7146  xpfir  7267  infxp  8028  infmap2  8031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1st 6288  df-2nd 6289  df-en 7046
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