Theorem xpmapen 6828

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 5496	. . . 4 |- ( w = z -> ( x ` w ) = ( x ` z ) )
5	4	fveq2d 5500	. . 3 |- ( w = z -> ( 1st ` ( x ` w ) ) = ( 1st ` ( x ` z ) ) )
6	5	cbvmptv 4085	. 2 |- ( w e. C |-> ( 1st ` ( x ` w ) ) ) = ( z e. C |-> ( 1st ` ( x ` z ) ) )
7	4	fveq2d 5500	. . 3 |- ( w = z -> ( 2nd ` ( x ` w ) ) = ( 2nd ` ( x ` z ) ) )
8	7	cbvmptv 4085	. 2 |- ( w e. C |-> ( 2nd ` ( x ` w ) ) ) = ( z e. C |-> ( 2nd ` ( x ` z ) ) )
9		fveq2 5496	. . . 4 |- ( w = z -> ( ( 1st ` y ) ` w ) = ( ( 1st ` y ) ` z ) )
10		fveq2 5496	. . . 4 |- ( w = z -> ( ( 2nd ` y ) ` w ) = ( ( 2nd ` y ) ` z ) )
11	9, 10	opeq12d 3773	. . 3 |- ( w = z -> <. ( ( 1st ` y ) ` w ) , ( ( 2nd ` y ) ` w ) >. = <. ( ( 1st ` y ) ` z ) , ( ( 2nd ` y ) ` z ) >. )
12	11	cbvmptv 4085	. 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 6827	1 |- ( ( A X. B ) ^m C ) ~~ ( ( A ^m C ) X. ( B ^m C ) )

Syntax hints:   e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   X. cxp 4609   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   ^m cmap 6626   ~~ cen 6716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-en 6719

This theorem is referenced by: (None)