Theorem fimaproj 28084

Description: Image of a cartesian product for a function on pairs, given two projections F and G. (Contributed by Thierry Arnoux, 30-Dec-2019.)

Hypotheses

Ref	Expression
fvproj.h	|- H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. )
fimaproj.f	|- ( ph -> F Fn A )
fimaproj.g	|- ( ph -> G Fn B )
fimaproj.x	|- ( ph -> X C_ A )
fimaproj.y	|- ( ph -> Y C_ B )

Assertion

Ref	Expression
fimaproj	|- ( ph -> ( H " ( X X. Y ) ) = ( ( F " X ) X. ( G " Y ) ) )

Distinct variable groups:   x, A, y   x, B, y   x, F, y   x, G, y   x, H, y

Allowed substitution hints:   ph( x, y)   X( x, y)   Y( x, y)

Proof of Theorem fimaproj

Dummy variables a b z c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 4720	. . . . 5 |- <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. e. _V
2		fvproj.h	. . . . . 6 |- H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. )
3		vex 3112	. . . . . . . . . 10 |- x e. _V
4		vex 3112	. . . . . . . . . 10 |- y e. _V
5	3, 4	op1std 6809	. . . . . . . . 9 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
6	5	fveq2d 5876	. . . . . . . 8 |- ( z = <. x , y >. -> ( F ` ( 1st ` z ) ) = ( F ` x ) )
7	3, 4	op2ndd 6810	. . . . . . . . 9 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
8	7	fveq2d 5876	. . . . . . . 8 |- ( z = <. x , y >. -> ( G ` ( 2nd ` z ) ) = ( G ` y ) )
9	6, 8	opeq12d 4227	. . . . . . 7 |- ( z = <. x , y >. -> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. = <. ( F ` x ) , ( G ` y ) >. )
10	9	mpt2mpt 6393	. . . . . 6 |- ( z e. ( A X. B ) |-> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. ) = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. )
11	2, 10	eqtr4i 2489	. . . . 5 |- H = ( z e. ( A X. B ) |-> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. )
12	1, 11	fnmpti 5715	. . . 4 |- H Fn ( A X. B )
13		fimaproj.x	. . . . 5 |- ( ph -> X C_ A )
14		fimaproj.y	. . . . 5 |- ( ph -> Y C_ B )
15		xpss12 5117	. . . . 5 |- ( ( X C_ A /\ Y C_ B ) -> ( X X. Y ) C_ ( A X. B ) )
16	13, 14, 15	syl2anc 661	. . . 4 |- ( ph -> ( X X. Y ) C_ ( A X. B ) )
17		fvelimab 5929	. . . 4 |- ( ( H Fn ( A X. B ) /\ ( X X. Y ) C_ ( A X. B ) ) -> ( c e. ( H " ( X X. Y ) ) <-> E. z e. ( X X. Y ) ( H ` z ) = c ) )
18	12, 16, 17	sylancr 663	. . 3 |- ( ph -> ( c e. ( H " ( X X. Y ) ) <-> E. z e. ( X X. Y ) ( H ` z ) = c ) )
19		simp-4r 768	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> a e. X )
20		simplr 755	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> b e. Y )
21		opelxpi 5040	. . . . . . . 8 |- ( ( a e. X /\ b e. Y ) -> <. a , b >. e. ( X X. Y ) )
22	19, 20, 21	syl2anc 661	. . . . . . 7 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> <. a , b >. e. ( X X. Y ) )
23		simpllr 760	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> ( F ` a ) = ( 1st ` c ) )
24		simpr 461	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> ( G ` b ) = ( 2nd ` c ) )
25	23, 24	opeq12d 4227	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> <. ( F ` a ) , ( G ` b ) >. = <. ( 1st ` c ) , ( 2nd ` c ) >. )
26	13	ad5antr 733	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> X C_ A )
27	26, 19	sseldd 3500	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> a e. A )
28	14	ad5antr 733	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> Y C_ B )
29	28, 20	sseldd 3500	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> b e. B )
30	2, 27, 29	fvproj 28083	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> ( H ` <. a , b >. ) = <. ( F ` a ) , ( G ` b ) >. )
31		1st2nd2 6836	. . . . . . . . 9 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> c = <. ( 1st ` c ) , ( 2nd ` c ) >. )
32	31	ad5antlr 734	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> c = <. ( 1st ` c ) , ( 2nd ` c ) >. )
33	25, 30, 32	3eqtr4d 2508	. . . . . . 7 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> ( H ` <. a , b >. ) = c )
34		fveq2 5872	. . . . . . . . 9 |- ( z = <. a , b >. -> ( H ` z ) = ( H ` <. a , b >. ) )
35	34	eqeq1d 2459	. . . . . . . 8 |- ( z = <. a , b >. -> ( ( H ` z ) = c <-> ( H ` <. a , b >. ) = c ) )
36	35	rspcev 3210	. . . . . . 7 |- ( ( <. a , b >. e. ( X X. Y ) /\ ( H ` <. a , b >. ) = c ) -> E. z e. ( X X. Y ) ( H ` z ) = c )
37	22, 33, 36	syl2anc 661	. . . . . 6 |- ( ( ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) /\ b e. Y ) /\ ( G ` b ) = ( 2nd ` c ) ) -> E. z e. ( X X. Y ) ( H ` z ) = c )
38		fimaproj.g	. . . . . . . . 9 |- ( ph -> G Fn B )
39	38	ad3antrrr 729	. . . . . . . 8 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> G Fn B )
40		fnfun 5684	. . . . . . . 8 |- ( G Fn B -> Fun G )
41	39, 40	syl 16	. . . . . . 7 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> Fun G )
42		xp2nd 6830	. . . . . . . 8 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> ( 2nd ` c ) e. ( G " Y ) )
43	42	ad3antlr 730	. . . . . . 7 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> ( 2nd ` c ) e. ( G " Y ) )
44		fvelima 5925	. . . . . . 7 |- ( ( Fun G /\ ( 2nd ` c ) e. ( G " Y ) ) -> E. b e. Y ( G ` b ) = ( 2nd ` c ) )
45	41, 43, 44	syl2anc 661	. . . . . 6 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> E. b e. Y ( G ` b ) = ( 2nd ` c ) )
46	37, 45	r19.29a 2999	. . . . 5 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> E. z e. ( X X. Y ) ( H ` z ) = c )
47		fimaproj.f	. . . . . . . 8 |- ( ph -> F Fn A )
48	47	adantr 465	. . . . . . 7 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> F Fn A )
49		fnfun 5684	. . . . . . 7 |- ( F Fn A -> Fun F )
50	48, 49	syl 16	. . . . . 6 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> Fun F )
51		xp1st 6829	. . . . . . 7 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> ( 1st ` c ) e. ( F " X ) )
52	51	adantl 466	. . . . . 6 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> ( 1st ` c ) e. ( F " X ) )
53		fvelima 5925	. . . . . 6 |- ( ( Fun F /\ ( 1st ` c ) e. ( F " X ) ) -> E. a e. X ( F ` a ) = ( 1st ` c ) )
54	50, 52, 53	syl2anc 661	. . . . 5 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> E. a e. X ( F ` a ) = ( 1st ` c ) )
55	46, 54	r19.29a 2999	. . . 4 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> E. z e. ( X X. Y ) ( H ` z ) = c )
56		simpr 461	. . . . . 6 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) = c )
57	16	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( X X. Y ) C_ ( A X. B ) )
58		simplr 755	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> z e. ( X X. Y ) )
59	57, 58	sseldd 3500	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> z e. ( A X. B ) )
60	11	fvmpt2 5964	. . . . . . . 8 |- ( ( z e. ( A X. B ) /\ <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. e. _V ) -> ( H ` z ) = <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. )
61	59, 1, 60	sylancl 662	. . . . . . 7 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) = <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. )
62	47	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> F Fn A )
63	13	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> X C_ A )
64		xp1st 6829	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
65	58, 64	syl 16	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( 1st ` z ) e. X )
66		fnfvima 6151	. . . . . . . . 9 |- ( ( F Fn A /\ X C_ A /\ ( 1st ` z ) e. X ) -> ( F ` ( 1st ` z ) ) e. ( F " X ) )
67	62, 63, 65, 66	syl3anc 1228	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( F ` ( 1st ` z ) ) e. ( F " X ) )
68	38	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> G Fn B )
69	14	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> Y C_ B )
70		xp2nd 6830	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
71	58, 70	syl 16	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( 2nd ` z ) e. Y )
72		fnfvima 6151	. . . . . . . . 9 |- ( ( G Fn B /\ Y C_ B /\ ( 2nd ` z ) e. Y ) -> ( G ` ( 2nd ` z ) ) e. ( G " Y ) )
73	68, 69, 71, 72	syl3anc 1228	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( G ` ( 2nd ` z ) ) e. ( G " Y ) )
74		opelxpi 5040	. . . . . . . 8 |- ( ( ( F ` ( 1st ` z ) ) e. ( F " X ) /\ ( G ` ( 2nd ` z ) ) e. ( G " Y ) ) -> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. e. ( ( F " X ) X. ( G " Y ) ) )
75	67, 73, 74	syl2anc 661	. . . . . . 7 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. e. ( ( F " X ) X. ( G " Y ) ) )
76	61, 75	eqeltrd 2545	. . . . . 6 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) e. ( ( F " X ) X. ( G " Y ) ) )
77	56, 76	eqeltrrd 2546	. . . . 5 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> c e. ( ( F " X ) X. ( G " Y ) ) )
78	77	r19.29an 2998	. . . 4 |- ( ( ph /\ E. z e. ( X X. Y ) ( H ` z ) = c ) -> c e. ( ( F " X ) X. ( G " Y ) ) )
79	55, 78	impbida 832	. . 3 |- ( ph -> ( c e. ( ( F " X ) X. ( G " Y ) ) <-> E. z e. ( X X. Y ) ( H ` z ) = c ) )
80	18, 79	bitr4d 256	. 2 |- ( ph -> ( c e. ( H " ( X X. Y ) ) <-> c e. ( ( F " X ) X. ( G " Y ) ) ) )
81	80	eqrdv 2454	1 |- ( ph -> ( H " ( X X. Y ) ) = ( ( F " X ) X. ( G " Y ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   E.wrex 2808   _Vcvv 3109   C_ wss 3471   <.cop 4038   |-> cmpt 4515   X. cxp 5006   "cima 5011   Fun wfun 5588   Fn wfn 5589   ` cfv 5594   |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800

This theorem is referenced by:  txomap  28085