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Theorem xpmapen 7590
Description: Equinumerosity law for set exponentiation of a Cartesian 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 5800 . . . 4  |-  ( w  =  z  ->  (
x `  w )  =  ( x `  z ) )
54fveq2d 5804 . . 3  |-  ( w  =  z  ->  ( 1st `  ( x `  w ) )  =  ( 1st `  (
x `  z )
) )
65cbvmptv 4492 . 2  |-  ( w  e.  C  |->  ( 1st `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 1st `  (
x `  z )
) )
74fveq2d 5804 . . 3  |-  ( w  =  z  ->  ( 2nd `  ( x `  w ) )  =  ( 2nd `  (
x `  z )
) )
87cbvmptv 4492 . 2  |-  ( w  e.  C  |->  ( 2nd `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 2nd `  (
x `  z )
) )
9 fveq2 5800 . . . 4  |-  ( w  =  z  ->  (
( 1st `  y
) `  w )  =  ( ( 1st `  y ) `  z
) )
10 fveq2 5800 . . . 4  |-  ( w  =  z  ->  (
( 2nd `  y
) `  w )  =  ( ( 2nd `  y ) `  z
) )
119, 10opeq12d 4176 . . 3  |-  ( w  =  z  ->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >.  =  <. (
( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
1211cbvmptv 4492 . 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 7589 1  |-  ( ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   _Vcvv 3078   <.cop 3992   class class class wbr 4401    |-> cmpt 4459    X. cxp 4947   ` cfv 5527  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687    ^m cmap 7325    ~~ cen 7418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-en 7422
This theorem is referenced by:  rexpen  13629
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