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Theorem xpmapen 7029
Description: Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
xpmapen.1  |-  A  e. 
_V
xpmapen.2  |-  B  e. 
_V
xpmapen.3  |-  C  e. 
_V
Assertion
Ref Expression
xpmapen  |-  ( ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )

Proof of Theorem xpmapen
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmapen.1 . 2  |-  A  e. 
_V
2 xpmapen.2 . 2  |-  B  e. 
_V
3 xpmapen.3 . 2  |-  C  e. 
_V
4 fveq2 5525 . . . 4  |-  ( w  =  z  ->  (
x `  w )  =  ( x `  z ) )
54fveq2d 5529 . . 3  |-  ( w  =  z  ->  ( 1st `  ( x `  w ) )  =  ( 1st `  (
x `  z )
) )
65cbvmptv 4111 . 2  |-  ( w  e.  C  |->  ( 1st `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 1st `  (
x `  z )
) )
74fveq2d 5529 . . 3  |-  ( w  =  z  ->  ( 2nd `  ( x `  w ) )  =  ( 2nd `  (
x `  z )
) )
87cbvmptv 4111 . 2  |-  ( w  e.  C  |->  ( 2nd `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 2nd `  (
x `  z )
) )
9 fveq2 5525 . . . 4  |-  ( w  =  z  ->  (
( 1st `  y
) `  w )  =  ( ( 1st `  y ) `  z
) )
10 fveq2 5525 . . . 4  |-  ( w  =  z  ->  (
( 2nd `  y
) `  w )  =  ( ( 2nd `  y ) `  z
) )
119, 10opeq12d 3804 . . 3  |-  ( w  =  z  ->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >.  =  <. (
( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
1211cbvmptv 4111 . 2  |-  ( w  e.  C  |->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >. )  =  ( z  e.  C  |->  <.
( ( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
131, 2, 3, 6, 8, 12xpmapenlem 7028 1  |-  ( ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772    ~~ cen 6860
This theorem is referenced by:  rexpen  12506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-en 6864
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