Theorem xpmapen 7011

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 5627	. . . 4 |- ( w = z -> ( x ` w ) = ( x ` z ) )
5	4	fveq2d 5631	. . 3 |- ( w = z -> ( 1st ` ( x ` w ) ) = ( 1st ` ( x ` z ) ) )
6	5	cbvmptv 4180	. 2 |- ( w e. C |-> ( 1st ` ( x ` w ) ) ) = ( z e. C |-> ( 1st ` ( x ` z ) ) )
7	4	fveq2d 5631	. . 3 |- ( w = z -> ( 2nd ` ( x ` w ) ) = ( 2nd ` ( x ` z ) ) )
8	7	cbvmptv 4180	. 2 |- ( w e. C |-> ( 2nd ` ( x ` w ) ) ) = ( z e. C |-> ( 2nd ` ( x ` z ) ) )
9		fveq2 5627	. . . 4 |- ( w = z -> ( ( 1st ` y ) ` w ) = ( ( 1st ` y ) ` z ) )
10		fveq2 5627	. . . 4 |- ( w = z -> ( ( 2nd ` y ) ` w ) = ( ( 2nd ` y ) ` z ) )
11	9, 10	opeq12d 3865	. . 3 |- ( w = z -> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. = <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
12	11	cbvmptv 4180	. 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 7010	1 |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )

Syntax hints:   e. wcel 2200   _Vcvv 2799   <.cop 3669   class class class wbr 4083   |-> cmpt 4145   X. cxp 4717   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   ^m cmap 6795   ~~ cen 6885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-en 6888

This theorem is referenced by: (None)