Theorem xppreima 29449

Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)

Assertion

Ref	Expression
xppreima	|- ( ( Fun F /\ ran F C_ ( _V X. _V ) ) -> ( `' F " ( Y X. Z ) ) = ( ( `' ( 1st o. F ) " Y ) i^i ( `' ( 2nd o. F ) " Z ) ) )

Proof of Theorem xppreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 5918	. . . . 5 |- ( Fun F <-> F Fn dom F )
2		fncnvima2 6339	. . . . 5 |- ( F Fn dom F -> ( `' F " ( Y X. Z ) ) = { x e. dom F | ( F ` x ) e. ( Y X. Z ) } )
3	1, 2	sylbi 207	. . . 4 |- ( Fun F -> ( `' F " ( Y X. Z ) ) = { x e. dom F | ( F ` x ) e. ( Y X. Z ) } )
4	3	adantr 481	. . 3 |- ( ( Fun F /\ ran F C_ ( _V X. _V ) ) -> ( `' F " ( Y X. Z ) ) = { x e. dom F | ( F ` x ) e. ( Y X. Z ) } )
5		fvco 6274	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( 1st o. F ) ` x ) = ( 1st ` ( F ` x ) ) )
6		fvco 6274	. . . . . . . . . 10 |- ( ( Fun F /\ x e. dom F ) -> ( ( 2nd o. F ) ` x ) = ( 2nd ` ( F ` x ) ) )
7	5, 6	opeq12d 4410	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. )
8	7	eqeq2d 2632	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. <-> ( F ` x ) = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. ) )
9	5	eleq1d 2686	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( 1st o. F ) ` x ) e. Y <-> ( 1st ` ( F ` x ) ) e. Y ) )
10	6	eleq1d 2686	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( 2nd o. F ) ` x ) e. Z <-> ( 2nd ` ( F ` x ) ) e. Z ) )
11	9, 10	anbi12d 747	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) <-> ( ( 1st ` ( F ` x ) ) e. Y /\ ( 2nd ` ( F ` x ) ) e. Z ) ) )
12	8, 11	anbi12d 747	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. /\ ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) ) <-> ( ( F ` x ) = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. /\ ( ( 1st ` ( F ` x ) ) e. Y /\ ( 2nd ` ( F ` x ) ) e. Z ) ) ) )
13		elxp6 7200	. . . . . . 7 |- ( ( F ` x ) e. ( Y X. Z ) <-> ( ( F ` x ) = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. /\ ( ( 1st ` ( F ` x ) ) e. Y /\ ( 2nd ` ( F ` x ) ) e. Z ) ) )
14	12, 13	syl6rbbr 279	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. ( Y X. Z ) <-> ( ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. /\ ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) ) ) )
15	14	adantlr 751	. . . . 5 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( ( F ` x ) e. ( Y X. Z ) <-> ( ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. /\ ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) ) ) )
16		opfv 29448	. . . . . 6 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. )
17	16	biantrurd 529	. . . . 5 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) <-> ( ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. /\ ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) ) ) )
18		fo1st 7188	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
19		fofun 6116	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> Fun 1st )
20	18, 19	ax-mp 5	. . . . . . . . . 10 |- Fun 1st
21		funco 5928	. . . . . . . . . 10 |- ( ( Fun 1st /\ Fun F ) -> Fun ( 1st o. F ) )
22	20, 21	mpan 706	. . . . . . . . 9 |- ( Fun F -> Fun ( 1st o. F ) )
23	22	adantr 481	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> Fun ( 1st o. F ) )
24		ssv 3625	. . . . . . . . . . . 12 |- ( F " dom F ) C_ _V
25		fof 6115	. . . . . . . . . . . . 13 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
26		fdm 6051	. . . . . . . . . . . . 13 |- ( 1st : _V --> _V -> dom 1st = _V )
27	18, 25, 26	mp2b 10	. . . . . . . . . . . 12 |- dom 1st = _V
28	24, 27	sseqtr4i 3638	. . . . . . . . . . 11 |- ( F " dom F ) C_ dom 1st
29		ssid 3624	. . . . . . . . . . . 12 |- dom F C_ dom F
30		funimass3 6333	. . . . . . . . . . . 12 |- ( ( Fun F /\ dom F C_ dom F ) -> ( ( F " dom F ) C_ dom 1st <-> dom F C_ ( `' F " dom 1st ) ) )
31	29, 30	mpan2 707	. . . . . . . . . . 11 |- ( Fun F -> ( ( F " dom F ) C_ dom 1st <-> dom F C_ ( `' F " dom 1st ) ) )
32	28, 31	mpbii 223	. . . . . . . . . 10 |- ( Fun F -> dom F C_ ( `' F " dom 1st ) )
33	32	sselda 3603	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> x e. ( `' F " dom 1st ) )
34		dmco 5643	. . . . . . . . 9 |- dom ( 1st o. F ) = ( `' F " dom 1st )
35	33, 34	syl6eleqr 2712	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> x e. dom ( 1st o. F ) )
36		fvimacnv 6332	. . . . . . . 8 |- ( ( Fun ( 1st o. F ) /\ x e. dom ( 1st o. F ) ) -> ( ( ( 1st o. F ) ` x ) e. Y <-> x e. ( `' ( 1st o. F ) " Y ) ) )
37	23, 35, 36	syl2anc 693	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( 1st o. F ) ` x ) e. Y <-> x e. ( `' ( 1st o. F ) " Y ) ) )
38		fo2nd 7189	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
39		fofun 6116	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
40	38, 39	ax-mp 5	. . . . . . . . . 10 |- Fun 2nd
41		funco 5928	. . . . . . . . . 10 |- ( ( Fun 2nd /\ Fun F ) -> Fun ( 2nd o. F ) )
42	40, 41	mpan 706	. . . . . . . . 9 |- ( Fun F -> Fun ( 2nd o. F ) )
43	42	adantr 481	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> Fun ( 2nd o. F ) )
44		fof 6115	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
45		fdm 6051	. . . . . . . . . . . . 13 |- ( 2nd : _V --> _V -> dom 2nd = _V )
46	38, 44, 45	mp2b 10	. . . . . . . . . . . 12 |- dom 2nd = _V
47	24, 46	sseqtr4i 3638	. . . . . . . . . . 11 |- ( F " dom F ) C_ dom 2nd
48		funimass3 6333	. . . . . . . . . . . 12 |- ( ( Fun F /\ dom F C_ dom F ) -> ( ( F " dom F ) C_ dom 2nd <-> dom F C_ ( `' F " dom 2nd ) ) )
49	29, 48	mpan2 707	. . . . . . . . . . 11 |- ( Fun F -> ( ( F " dom F ) C_ dom 2nd <-> dom F C_ ( `' F " dom 2nd ) ) )
50	47, 49	mpbii 223	. . . . . . . . . 10 |- ( Fun F -> dom F C_ ( `' F " dom 2nd ) )
51	50	sselda 3603	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> x e. ( `' F " dom 2nd ) )
52		dmco 5643	. . . . . . . . 9 |- dom ( 2nd o. F ) = ( `' F " dom 2nd )
53	51, 52	syl6eleqr 2712	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> x e. dom ( 2nd o. F ) )
54		fvimacnv 6332	. . . . . . . 8 |- ( ( Fun ( 2nd o. F ) /\ x e. dom ( 2nd o. F ) ) -> ( ( ( 2nd o. F ) ` x ) e. Z <-> x e. ( `' ( 2nd o. F ) " Z ) ) )
55	43, 53, 54	syl2anc 693	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( 2nd o. F ) ` x ) e. Z <-> x e. ( `' ( 2nd o. F ) " Z ) ) )
56	37, 55	anbi12d 747	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) <-> ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) ) )
57	56	adantlr 751	. . . . 5 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( ( ( ( 1st o. F ) ` x ) e. Y /\ ( ( 2nd o. F ) ` x ) e. Z ) <-> ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) ) )
58	15, 17, 57	3bitr2d 296	. . . 4 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( ( F ` x ) e. ( Y X. Z ) <-> ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) ) )
59	58	rabbidva 3188	. . 3 |- ( ( Fun F /\ ran F C_ ( _V X. _V ) ) -> { x e. dom F | ( F ` x ) e. ( Y X. Z ) } = { x e. dom F | ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) } )
60	4, 59	eqtrd 2656	. 2 |- ( ( Fun F /\ ran F C_ ( _V X. _V ) ) -> ( `' F " ( Y X. Z ) ) = { x e. dom F | ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) } )
61		dfin5 3582	. . . 4 |- ( dom F i^i ( `' ( 1st o. F ) " Y ) ) = { x e. dom F | x e. ( `' ( 1st o. F ) " Y ) }
62		dfin5 3582	. . . 4 |- ( dom F i^i ( `' ( 2nd o. F ) " Z ) ) = { x e. dom F | x e. ( `' ( 2nd o. F ) " Z ) }
63	61, 62	ineq12i 3812	. . 3 |- ( ( dom F i^i ( `' ( 1st o. F ) " Y ) ) i^i ( dom F i^i ( `' ( 2nd o. F ) " Z ) ) ) = ( { x e. dom F | x e. ( `' ( 1st o. F ) " Y ) } i^i { x e. dom F | x e. ( `' ( 2nd o. F ) " Z ) } )
64		cnvimass 5485	. . . . . 6 |- ( `' ( 1st o. F ) " Y ) C_ dom ( 1st o. F )
65		dmcoss 5385	. . . . . 6 |- dom ( 1st o. F ) C_ dom F
66	64, 65	sstri 3612	. . . . 5 |- ( `' ( 1st o. F ) " Y ) C_ dom F
67		sseqin2 3817	. . . . 5 |- ( ( `' ( 1st o. F ) " Y ) C_ dom F <-> ( dom F i^i ( `' ( 1st o. F ) " Y ) ) = ( `' ( 1st o. F ) " Y ) )
68	66, 67	mpbi 220	. . . 4 |- ( dom F i^i ( `' ( 1st o. F ) " Y ) ) = ( `' ( 1st o. F ) " Y )
69		cnvimass 5485	. . . . . 6 |- ( `' ( 2nd o. F ) " Z ) C_ dom ( 2nd o. F )
70		dmcoss 5385	. . . . . 6 |- dom ( 2nd o. F ) C_ dom F
71	69, 70	sstri 3612	. . . . 5 |- ( `' ( 2nd o. F ) " Z ) C_ dom F
72		sseqin2 3817	. . . . 5 |- ( ( `' ( 2nd o. F ) " Z ) C_ dom F <-> ( dom F i^i ( `' ( 2nd o. F ) " Z ) ) = ( `' ( 2nd o. F ) " Z ) )
73	71, 72	mpbi 220	. . . 4 |- ( dom F i^i ( `' ( 2nd o. F ) " Z ) ) = ( `' ( 2nd o. F ) " Z )
74	68, 73	ineq12i 3812	. . 3 |- ( ( dom F i^i ( `' ( 1st o. F ) " Y ) ) i^i ( dom F i^i ( `' ( 2nd o. F ) " Z ) ) ) = ( ( `' ( 1st o. F ) " Y ) i^i ( `' ( 2nd o. F ) " Z ) )
75		inrab 3899	. . 3 |- ( { x e. dom F | x e. ( `' ( 1st o. F ) " Y ) } i^i { x e. dom F | x e. ( `' ( 2nd o. F ) " Z ) } ) = { x e. dom F | ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) }
76	63, 74, 75	3eqtr3ri 2653	. 2 |- { x e. dom F | ( x e. ( `' ( 1st o. F ) " Y ) /\ x e. ( `' ( 2nd o. F ) " Z ) ) } = ( ( `' ( 1st o. F ) " Y ) i^i ( `' ( 2nd o. F ) " Z ) )
77	60, 76	syl6eq 2672	1 |- ( ( Fun F /\ ran F C_ ( _V X. _V ) ) -> ( `' F " ( Y X. Z ) ) = ( ( `' ( 1st o. F ) " Y ) i^i ( `' ( 2nd o. F ) " Z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  xppreima2  29450