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Theorem xpmapen 6997
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
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 5458 . . . 4  |-  ( w  =  z  ->  (
x `  w )  =  ( x `  z ) )
54fveq2d 5462 . . 3  |-  ( w  =  z  ->  ( 1st `  ( x `  w ) )  =  ( 1st `  (
x `  z )
) )
65cbvmptv 4085 . 2  |-  ( w  e.  C  |->  ( 1st `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 1st `  (
x `  z )
) )
74fveq2d 5462 . . 3  |-  ( w  =  z  ->  ( 2nd `  ( x `  w ) )  =  ( 2nd `  (
x `  z )
) )
87cbvmptv 4085 . 2  |-  ( w  e.  C  |->  ( 2nd `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 2nd `  (
x `  z )
) )
9 fveq2 5458 . . . 4  |-  ( w  =  z  ->  (
( 1st `  y
) `  w )  =  ( ( 1st `  y ) `  z
) )
10 fveq2 5458 . . . 4  |-  ( w  =  z  ->  (
( 2nd `  y
) `  w )  =  ( ( 2nd `  y ) `  z
) )
119, 10opeq12d 3778 . . 3  |-  ( w  =  z  ->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >.  =  <. (
( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
1211cbvmptv 4085 . 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 6996 1  |-  ( ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2763   <.cop 3617   class class class wbr 3997    e. cmpt 4051    X. cxp 4659   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055    ^m cmap 6740    ~~ cen 6828
This theorem is referenced by:  rexpen  12468
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-map 6742  df-en 6832
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