Theorem xpmapen 7140

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.

Step	Hyp	Ref	Expression
1		xpmapen.1	. 2 |- A e. _V
2		xpmapen.2	. 2 |- B e. _V
3		xpmapen.3	. 2 |- C e. _V
4		fveq2 5690	. . . 4 |- ( w = z -> ( x ` w ) = ( x ` z ) )
5	4	fveq2d 5694	. . 3 |- ( w = z -> ( 1st ` ( x ` w ) ) = ( 1st ` ( x ` z ) ) )
6	5	cbvmptv 4222	. 2 |- ( w e. C |-> ( 1st ` ( x ` w ) ) ) = ( z e. C |-> ( 1st ` ( x ` z ) ) )
7	4	fveq2d 5694	. . 3 |- ( w = z -> ( 2nd ` ( x ` w ) ) = ( 2nd ` ( x ` z ) ) )
8	7	cbvmptv 4222	. 2 |- ( w e. C |-> ( 2nd ` ( x ` w ) ) ) = ( z e. C |-> ( 2nd ` ( x ` z ) ) )
9		fveq2 5690	. . . 4 |- ( w = z -> ( ( 1st ` y ) ` w ) = ( ( 1st ` y ) ` z ) )
10		fveq2 5690	. . . 4 |- ( w = z -> ( ( 2nd ` y ) ` w ) = ( ( 2nd ` y ) ` z ) )
11	9, 10	opeq12d 3907	. . 3 |- ( w = z -> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. = <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
12	11	cbvmptv 4222	. 2 |- ( w e. C |-> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. ) = ( z e. C |-> <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
13	1, 2, 3, 6, 8, 12	xpmapenlem 7139	1 |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )

Syntax hints:   e. wcel 2209   _Vcvv 2821   <.cop 3708   class class class wbr 4125   |-> cmpt 4187   X. cxp 4767   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   ^m cmap 6912   ~~ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-en 7013

This theorem is referenced by: (None)