Theorem xpmapen 6920

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 5561	. . . 4 |- ( w = z -> ( x ` w ) = ( x ` z ) )
5	4	fveq2d 5565	. . 3 |- ( w = z -> ( 1st ` ( x ` w ) ) = ( 1st ` ( x ` z ) ) )
6	5	cbvmptv 4130	. 2 |- ( w e. C |-> ( 1st ` ( x ` w ) ) ) = ( z e. C |-> ( 1st ` ( x ` z ) ) )
7	4	fveq2d 5565	. . 3 |- ( w = z -> ( 2nd ` ( x ` w ) ) = ( 2nd ` ( x ` z ) ) )
8	7	cbvmptv 4130	. 2 |- ( w e. C |-> ( 2nd ` ( x ` w ) ) ) = ( z e. C |-> ( 2nd ` ( x ` z ) ) )
9		fveq2 5561	. . . 4 |- ( w = z -> ( ( 1st ` y ) ` w ) = ( ( 1st ` y ) ` z ) )
10		fveq2 5561	. . . 4 |- ( w = z -> ( ( 2nd ` y ) ` w ) = ( ( 2nd ` y ) ` z ) )
11	9, 10	opeq12d 3817	. . 3 |- ( w = z -> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. = <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
12	11	cbvmptv 4130	. 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 6919	1 |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )

Syntax hints:   e. wcel 2167   _Vcvv 2763   <.cop 3626   class class class wbr 4034   |-> cmpt 4095   X. cxp 4662   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   ^m cmap 6716   ~~ cen 6806

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-en 6809

This theorem is referenced by: (None)