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

Theorem xpmapen 6914
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 5377 . . . 4  |-  ( w  =  z  ->  (
x `  w )  =  ( x `  z ) )
54fveq2d 5381 . . 3  |-  ( w  =  z  ->  ( 1st `  ( x `  w ) )  =  ( 1st `  (
x `  z )
) )
65cbvmptv 4008 . 2  |-  ( w  e.  C  |->  ( 1st `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 1st `  (
x `  z )
) )
74fveq2d 5381 . . 3  |-  ( w  =  z  ->  ( 2nd `  ( x `  w ) )  =  ( 2nd `  (
x `  z )
) )
87cbvmptv 4008 . 2  |-  ( w  e.  C  |->  ( 2nd `  ( x `  w
) ) )  =  ( z  e.  C  |->  ( 2nd `  (
x `  z )
) )
9 fveq2 5377 . . . 4  |-  ( w  =  z  ->  (
( 1st `  y
) `  w )  =  ( ( 1st `  y ) `  z
) )
10 fveq2 5377 . . . 4  |-  ( w  =  z  ->  (
( 2nd `  y
) `  w )  =  ( ( 2nd `  y ) `  z
) )
119, 10opeq12d 3704 . . 3  |-  ( w  =  z  ->  <. (
( 1st `  y
) `  w ) ,  ( ( 2nd `  y ) `  w
) >.  =  <. (
( 1st `  y
) `  z ) ,  ( ( 2nd `  y ) `  z
) >. )
1211cbvmptv 4008 . 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 6913 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 2727   <.cop 3547   class class class wbr 3920    e. cmpt 3974    X. cxp 4578   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973    ^m cmap 6658    ~~ cen 6746
This theorem is referenced by:  rexpen  12380
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-map 6660  df-en 6750
  Copyright terms: Public domain W3C validator