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Theorem xpmapen 7678
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 5848 . . . 4  |-  ( w  =  z  ->  (
x `  w )  =  ( x `  z ) )
54fveq2d 5852 . . 3  |-  ( w  =  z  ->  ( 1st `  ( x `  w ) )  =  ( 1st `  (
x `  z )
) )
65cbvmptv 4530 . 2  |-  ( w  e.  C  |->  ( 1st `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 1st `  (
x `  z )
) )
74fveq2d 5852 . . 3  |-  ( w  =  z  ->  ( 2nd `  ( x `  w ) )  =  ( 2nd `  (
x `  z )
) )
87cbvmptv 4530 . 2  |-  ( w  e.  C  |->  ( 2nd `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 2nd `  (
x `  z )
) )
9 fveq2 5848 . . . 4  |-  ( w  =  z  ->  (
( 1st `  y
) `  w )  =  ( ( 1st `  y ) `  z
) )
10 fveq2 5848 . . . 4  |-  ( w  =  z  ->  (
( 2nd `  y
) `  w )  =  ( ( 2nd `  y ) `  z
) )
119, 10opeq12d 4211 . . 3  |-  ( w  =  z  ->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >.  =  <. (
( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
1211cbvmptv 4530 . 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 7677 1  |-  ( ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772    ^m cmap 7412    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-en 7510
This theorem is referenced by:  rexpen  14045
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