Theorem imasaddvallem 16189

Description: The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
imasaddf.f	|- ( ph -> F : V -onto-> B )
imasaddf.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
imasaddflem.a	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )

Assertion

Ref	Expression
imasaddvallem	|- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )

Distinct variable groups:   q, p, B   a, b, p, q, V   .x. , p, q   X, p   F, a, b, p, q   ph, a, b, p, q   .xb , a, b, p, q   Y, p, q

Allowed substitution hints:   B( a, b)   .x. ( a, b)   X( q, a, b)   Y( a, b)

Proof of Theorem imasaddvallem

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( ( F ` X ) .xb ( F ` Y ) ) = ( .xb ` <. ( F ` X ) , ( F ` Y ) >. )
2		imasaddf.f	. . . . . 6 |- ( ph -> F : V -onto-> B )
3		imasaddf.e	. . . . . 6 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
4		imasaddflem.a	. . . . . 6 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
5	2, 3, 4	imasaddfnlem 16188	. . . . 5 |- ( ph -> .xb Fn ( B X. B ) )
6		fnfun 5988	. . . . 5 |- ( .xb Fn ( B X. B ) -> Fun .xb )
7	5, 6	syl 17	. . . 4 |- ( ph -> Fun .xb )
8	7	3ad2ant1 1082	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> Fun .xb )
9		fveq2 6191	. . . . . . . . . . 11 |- ( p = X -> ( F ` p ) = ( F ` X ) )
10	9	opeq1d 4408	. . . . . . . . . 10 |- ( p = X -> <. ( F ` p ) , ( F ` Y ) >. = <. ( F ` X ) , ( F ` Y ) >. )
11		oveq1 6657	. . . . . . . . . . 11 |- ( p = X -> ( p .x. Y ) = ( X .x. Y ) )
12	11	fveq2d 6195	. . . . . . . . . 10 |- ( p = X -> ( F ` ( p .x. Y ) ) = ( F ` ( X .x. Y ) ) )
13	10, 12	opeq12d 4410	. . . . . . . . 9 |- ( p = X -> <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. = <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. )
14	13	sneqd 4189	. . . . . . . 8 |- ( p = X -> { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } = { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } )
15	14	ssiun2s 4564	. . . . . . 7 |- ( X e. V -> { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } C_ U_ p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } )
16	15	3ad2ant2 1083	. . . . . 6 |- ( ( ph /\ X e. V /\ Y e. V ) -> { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } C_ U_ p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } )
17		fveq2 6191	. . . . . . . . . . . . 13 |- ( q = Y -> ( F ` q ) = ( F ` Y ) )
18	17	opeq2d 4409	. . . . . . . . . . . 12 |- ( q = Y -> <. ( F ` p ) , ( F ` q ) >. = <. ( F ` p ) , ( F ` Y ) >. )
19		oveq2 6658	. . . . . . . . . . . . 13 |- ( q = Y -> ( p .x. q ) = ( p .x. Y ) )
20	19	fveq2d 6195	. . . . . . . . . . . 12 |- ( q = Y -> ( F ` ( p .x. q ) ) = ( F ` ( p .x. Y ) ) )
21	18, 20	opeq12d 4410	. . . . . . . . . . 11 |- ( q = Y -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. = <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. )
22	21	sneqd 4189	. . . . . . . . . 10 |- ( q = Y -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } = { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } )
23	22	ssiun2s 4564	. . . . . . . . 9 |- ( Y e. V -> { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
24	23	ralrimivw 2967	. . . . . . . 8 |- ( Y e. V -> A. p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
25		ss2iun 4536	. . . . . . . 8 |- ( A. p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } -> U_ p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
26	24, 25	syl 17	. . . . . . 7 |- ( Y e. V -> U_ p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
27	26	3ad2ant3 1084	. . . . . 6 |- ( ( ph /\ X e. V /\ Y e. V ) -> U_ p e. V { <. <. ( F ` p ) , ( F ` Y ) >. , ( F ` ( p .x. Y ) ) >. } C_ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
28	16, 27	sstrd 3613	. . . . 5 |- ( ( ph /\ X e. V /\ Y e. V ) -> { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } C_ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
29	4	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ X e. V /\ Y e. V ) -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
30	28, 29	sseqtr4d 3642	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } C_ .xb )
31		opex 4932	. . . . 5 |- <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. e. _V
32	31	snss 4316	. . . 4 |- ( <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. e. .xb <-> { <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. } C_ .xb )
33	30, 32	sylibr 224	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. e. .xb )
34		funopfv 6235	. . 3 |- ( Fun .xb -> ( <. <. ( F ` X ) , ( F ` Y ) >. , ( F ` ( X .x. Y ) ) >. e. .xb -> ( .xb ` <. ( F ` X ) , ( F ` Y ) >. ) = ( F ` ( X .x. Y ) ) ) )
35	8, 33, 34	sylc 65	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( .xb ` <. ( F ` X ) , ( F ` Y ) >. ) = ( F ` ( X .x. Y ) ) )
36	1, 35	syl5eq 2668	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653

This theorem is referenced by:  imasaddval  16192  imasmulval  16195  qusaddvallem  16211