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

Theorem xpcomeng 7192
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 4884 . . 3  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
2 xpeq2 4885 . . 3  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
31, 2breq12d 4217 . 2  |-  ( x  =  A  ->  (
( x  X.  y
)  ~~  ( y  X.  x )  <->  ( A  X.  y )  ~~  (
y  X.  A ) ) )
4 xpeq2 4885 . . 3  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
5 xpeq1 4884 . . 3  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
64, 5breq12d 4217 . 2  |-  ( y  =  B  ->  (
( A  X.  y
)  ~~  ( y  X.  A )  <->  ( A  X.  B )  ~~  ( B  X.  A ) ) )
7 vex 2951 . . 3  |-  x  e. 
_V
8 vex 2951 . . 3  |-  y  e. 
_V
97, 8xpcomen 7191 . 2  |-  ( x  X.  y )  ~~  ( y  X.  x
)
103, 6, 9vtocl2g 3007 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 1652    e. wcel 1725   class class class wbr 4204    X. cxp 4868    ~~ cen 7098
This theorem is referenced by:  xpsnen2g  7193  xpdom1g  7197  omxpen  7202  xpfir  7323  infxp  8087  infmap2  8090
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1st 6341  df-2nd 6342  df-en 7102
  Copyright terms: Public domain W3C validator