Theorem fimaproj 29900

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 4932	. . . . 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 3203	. . . . . . . . . 10 |- x e. _V
4		vex 3203	. . . . . . . . . 10 |- y e. _V
5	3, 4	op1std 7178	. . . . . . . . 9 |- ( z = <. x , y >. -> ( 1st ` z ) = x )
6	5	fveq2d 6195	. . . . . . . 8 |- ( z = <. x , y >. -> ( F ` ( 1st ` z ) ) = ( F ` x ) )
7	3, 4	op2ndd 7179	. . . . . . . . 9 |- ( z = <. x , y >. -> ( 2nd ` z ) = y )
8	7	fveq2d 6195	. . . . . . . 8 |- ( z = <. x , y >. -> ( G ` ( 2nd ` z ) ) = ( G ` y ) )
9	6, 8	opeq12d 4410	. . . . . . 7 |- ( z = <. x , y >. -> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. = <. ( F ` x ) , ( G ` y ) >. )
10	9	mpt2mpt 6752	. . . . . 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 2647	. . . . 5 |- H = ( z e. ( A X. B ) |-> <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. )
12	1, 11	fnmpti 6022	. . . 4 |- H Fn ( A X. B )
13		fimaproj.x	. . . . 5 |- ( ph -> X C_ A )
14		fimaproj.y	. . . . 5 |- ( ph -> Y C_ B )
15		xpss12 5225	. . . . 5 |- ( ( X C_ A /\ Y C_ B ) -> ( X X. Y ) C_ ( A X. B ) )
16	13, 14, 15	syl2anc 693	. . . 4 |- ( ph -> ( X X. Y ) C_ ( A X. B ) )
17		fvelimab 6253	. . . 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 695	. . 3 |- ( ph -> ( c e. ( H " ( X X. Y ) ) <-> E. z e. ( X X. Y ) ( H ` z ) = c ) )
19		simp-4r 807	. . . . . . . 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 792	. . . . . . . 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 5148	. . . . . . . 8 |- ( ( a e. X /\ b e. Y ) -> <. a , b >. e. ( X X. Y ) )
22	19, 20, 21	syl2anc 693	. . . . . . 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 799	. . . . . . . . 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 477	. . . . . . . . 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 4410	. . . . . . . 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 770	. . . . . . . . . 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 3604	. . . . . . . . 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 770	. . . . . . . . . 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 3604	. . . . . . . . 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 29899	. . . . . . . 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 7205	. . . . . . . . 9 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> c = <. ( 1st ` c ) , ( 2nd ` c ) >. )
32	31	ad5antlr 771	. . . . . . . 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 2666	. . . . . . 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 6191	. . . . . . . . 9 |- ( z = <. a , b >. -> ( H ` z ) = ( H ` <. a , b >. ) )
35	34	eqeq1d 2624	. . . . . . . 8 |- ( z = <. a , b >. -> ( ( H ` z ) = c <-> ( H ` <. a , b >. ) = c ) )
36	35	rspcev 3309	. . . . . . 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 693	. . . . . 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 766	. . . . . . . 8 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> G Fn B )
40		fnfun 5988	. . . . . . . 8 |- ( G Fn B -> Fun G )
41	39, 40	syl 17	. . . . . . 7 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> Fun G )
42		xp2nd 7199	. . . . . . . 8 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> ( 2nd ` c ) e. ( G " Y ) )
43	42	ad3antlr 767	. . . . . . 7 |- ( ( ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) /\ a e. X ) /\ ( F ` a ) = ( 1st ` c ) ) -> ( 2nd ` c ) e. ( G " Y ) )
44		fvelima 6248	. . . . . . 7 |- ( ( Fun G /\ ( 2nd ` c ) e. ( G " Y ) ) -> E. b e. Y ( G ` b ) = ( 2nd ` c ) )
45	41, 43, 44	syl2anc 693	. . . . . 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 3078	. . . . 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 481	. . . . . . 7 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> F Fn A )
49		fnfun 5988	. . . . . . 7 |- ( F Fn A -> Fun F )
50	48, 49	syl 17	. . . . . 6 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> Fun F )
51		xp1st 7198	. . . . . . 7 |- ( c e. ( ( F " X ) X. ( G " Y ) ) -> ( 1st ` c ) e. ( F " X ) )
52	51	adantl 482	. . . . . 6 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> ( 1st ` c ) e. ( F " X ) )
53		fvelima 6248	. . . . . 6 |- ( ( Fun F /\ ( 1st ` c ) e. ( F " X ) ) -> E. a e. X ( F ` a ) = ( 1st ` c ) )
54	50, 52, 53	syl2anc 693	. . . . 5 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> E. a e. X ( F ` a ) = ( 1st ` c ) )
55	46, 54	r19.29a 3078	. . . 4 |- ( ( ph /\ c e. ( ( F " X ) X. ( G " Y ) ) ) -> E. z e. ( X X. Y ) ( H ` z ) = c )
56		simpr 477	. . . . . 6 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) = c )
57	16	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( X X. Y ) C_ ( A X. B ) )
58		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> z e. ( X X. Y ) )
59	57, 58	sseldd 3604	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> z e. ( A X. B ) )
60	11	fvmpt2 6291	. . . . . . . 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 694	. . . . . . 7 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) = <. ( F ` ( 1st ` z ) ) , ( G ` ( 2nd ` z ) ) >. )
62	47	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> F Fn A )
63	13	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> X C_ A )
64		xp1st 7198	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X )
65	58, 64	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( 1st ` z ) e. X )
66		fnfvima 6496	. . . . . . . . 9 |- ( ( F Fn A /\ X C_ A /\ ( 1st ` z ) e. X ) -> ( F ` ( 1st ` z ) ) e. ( F " X ) )
67	62, 63, 65, 66	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( F ` ( 1st ` z ) ) e. ( F " X ) )
68	38	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> G Fn B )
69	14	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> Y C_ B )
70		xp2nd 7199	. . . . . . . . . 10 |- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y )
71	58, 70	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( 2nd ` z ) e. Y )
72		fnfvima 6496	. . . . . . . . 9 |- ( ( G Fn B /\ Y C_ B /\ ( 2nd ` z ) e. Y ) -> ( G ` ( 2nd ` z ) ) e. ( G " Y ) )
73	68, 69, 71, 72	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( G ` ( 2nd ` z ) ) e. ( G " Y ) )
74		opelxpi 5148	. . . . . . . 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 693	. . . . . . 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 2701	. . . . . 6 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> ( H ` z ) e. ( ( F " X ) X. ( G " Y ) ) )
77	56, 76	eqeltrrd 2702	. . . . 5 |- ( ( ( ph /\ z e. ( X X. Y ) ) /\ ( H ` z ) = c ) -> c e. ( ( F " X ) X. ( G " Y ) ) )
78	77	r19.29an 3077	. . . 4 |- ( ( ph /\ E. z e. ( X X. Y ) ( H ` z ) = c ) -> c e. ( ( F " X ) X. ( G " Y ) ) )
79	55, 78	impbida 877	. . 3 |- ( ph -> ( c e. ( ( F " X ) X. ( G " Y ) ) <-> E. z e. ( X X. Y ) ( H ` z ) = c ) )
80	18, 79	bitr4d 271	. 2 |- ( ph -> ( c e. ( H " ( X X. Y ) ) <-> c e. ( ( F " X ) X. ( G " Y ) ) ) )
81	80	eqrdv 2620	1 |- ( ph -> ( H " ( X X. Y ) ) = ( ( F " X ) X. ( G " Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   |-> cmpt 4729   X. cxp 5112   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  txomap  29901