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Metamath Proof Explorer

Theorem xpmapen

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)

Ref Expression
Hypotheses xpmapen.1 
|- A e. _V
xpmapen.2 
|- B e. _V
xpmapen.3 
|- C e. _V
Assertion xpmapen 
|- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )

Proof

Step Hyp Ref Expression
1 xpmapen.1 
 |-  A e. _V
2 xpmapen.2 
 |-  B e. _V
3 xpmapen.3 
 |-  C e. _V
4 2fveq3 
 |-  ( w = z -> ( 1st ` ( x ` w ) ) = ( 1st ` ( x ` z ) ) )
5 4 cbvmptv 
 |-  ( w e. C |-> ( 1st ` ( x ` w ) ) ) = ( z e. C |-> ( 1st ` ( x ` z ) ) )
6 2fveq3 
 |-  ( w = z -> ( 2nd ` ( x ` w ) ) = ( 2nd ` ( x ` z ) ) )
7 6 cbvmptv 
 |-  ( w e. C |-> ( 2nd ` ( x ` w ) ) ) = ( z e. C |-> ( 2nd ` ( x ` z ) ) )
8 fveq2 
 |-  ( w = z -> ( ( 1st ` y ) ` w ) = ( ( 1st ` y ) ` z ) )
9 fveq2 
 |-  ( w = z -> ( ( 2nd ` y ) ` w ) = ( ( 2nd ` y ) ` z ) )
10 8 9 opeq12d 
 |-  ( w = z -> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. = <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
11 10 cbvmptv 
 |-  ( w e. C |-> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. ) = ( z e. C |-> <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
12 1 2 3 5 7 11 xpmapenlem 
 |-  ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )