Theorem xppreima2 29450

Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)

Hypotheses

Ref	Expression
xppreima2.1	|- ( ph -> F : A --> B )
xppreima2.2	|- ( ph -> G : A --> C )
xppreima2.3	|- H = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. )

Assertion

Ref	Expression
xppreima2	|- ( ph -> ( `' H " ( Y X. Z ) ) = ( ( `' F " Y ) i^i ( `' G " Z ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, F   x, G   x, H   ph, x

Allowed substitution hints:   Y( x)   Z( x)

Proof of Theorem xppreima2

Step	Hyp	Ref	Expression
1		xppreima2.3	. . . 4 |- H = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. )
2	1	funmpt2 5927	. . 3 |- Fun H
3		xppreima2.1	. . . . . . . 8 |- ( ph -> F : A --> B )
4	3	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. B )
5		xppreima2.2	. . . . . . . 8 |- ( ph -> G : A --> C )
6	5	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. C )
7		opelxp 5146	. . . . . . 7 |- ( <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) <-> ( ( F ` x ) e. B /\ ( G ` x ) e. C ) )
8	4, 6, 7	sylanbrc 698	. . . . . 6 |- ( ( ph /\ x e. A ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
9	8, 1	fmptd 6385	. . . . 5 |- ( ph -> H : A --> ( B X. C ) )
10		frn 6053	. . . . 5 |- ( H : A --> ( B X. C ) -> ran H C_ ( B X. C ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ran H C_ ( B X. C ) )
12		xpss 5226	. . . 4 |- ( B X. C ) C_ ( _V X. _V )
13	11, 12	syl6ss 3615	. . 3 |- ( ph -> ran H C_ ( _V X. _V ) )
14		xppreima 29449	. . 3 |- ( ( Fun H /\ ran H C_ ( _V X. _V ) ) -> ( `' H " ( Y X. Z ) ) = ( ( `' ( 1st o. H ) " Y ) i^i ( `' ( 2nd o. H ) " Z ) ) )
15	2, 13, 14	sylancr 695	. 2 |- ( ph -> ( `' H " ( Y X. Z ) ) = ( ( `' ( 1st o. H ) " Y ) i^i ( `' ( 2nd o. H ) " Z ) ) )
16		fo1st 7188	. . . . . . . . 9 |- 1st : _V -onto-> _V
17		fofn 6117	. . . . . . . . 9 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
18	16, 17	ax-mp 5	. . . . . . . 8 |- 1st Fn _V
19		opex 4932	. . . . . . . . 9 |- <. ( F ` x ) , ( G ` x ) >. e. _V
20	19, 1	fnmpti 6022	. . . . . . . 8 |- H Fn A
21		ssv 3625	. . . . . . . 8 |- ran H C_ _V
22		fnco 5999	. . . . . . . 8 |- ( ( 1st Fn _V /\ H Fn A /\ ran H C_ _V ) -> ( 1st o. H ) Fn A )
23	18, 20, 21, 22	mp3an 1424	. . . . . . 7 |- ( 1st o. H ) Fn A
24	23	a1i 11	. . . . . 6 |- ( ph -> ( 1st o. H ) Fn A )
25		ffn 6045	. . . . . . 7 |- ( F : A --> B -> F Fn A )
26	3, 25	syl 17	. . . . . 6 |- ( ph -> F Fn A )
27	2	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> Fun H )
28	13	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ran H C_ ( _V X. _V ) )
29		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> x e. A )
30	19, 1	dmmpti 6023	. . . . . . . . . . 11 |- dom H = A
31	29, 30	syl6eleqr 2712	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. dom H )
32		opfv 29448	. . . . . . . . . 10 |- ( ( ( Fun H /\ ran H C_ ( _V X. _V ) ) /\ x e. dom H ) -> ( H ` x ) = <. ( ( 1st o. H ) ` x ) , ( ( 2nd o. H ) ` x ) >. )
33	27, 28, 31, 32	syl21anc 1325	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( H ` x ) = <. ( ( 1st o. H ) ` x ) , ( ( 2nd o. H ) ` x ) >. )
34	1	fvmpt2 6291	. . . . . . . . . 10 |- ( ( x e. A /\ <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) ) -> ( H ` x ) = <. ( F ` x ) , ( G ` x ) >. )
35	29, 8, 34	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( H ` x ) = <. ( F ` x ) , ( G ` x ) >. )
36	33, 35	eqtr3d 2658	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> <. ( ( 1st o. H ) ` x ) , ( ( 2nd o. H ) ` x ) >. = <. ( F ` x ) , ( G ` x ) >. )
37		fvex 6201	. . . . . . . . 9 |- ( ( 1st o. H ) ` x ) e. _V
38		fvex 6201	. . . . . . . . 9 |- ( ( 2nd o. H ) ` x ) e. _V
39	37, 38	opth 4945	. . . . . . . 8 |- ( <. ( ( 1st o. H ) ` x ) , ( ( 2nd o. H ) ` x ) >. = <. ( F ` x ) , ( G ` x ) >. <-> ( ( ( 1st o. H ) ` x ) = ( F ` x ) /\ ( ( 2nd o. H ) ` x ) = ( G ` x ) ) )
40	36, 39	sylib 208	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( ( 1st o. H ) ` x ) = ( F ` x ) /\ ( ( 2nd o. H ) ` x ) = ( G ` x ) ) )
41	40	simpld 475	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( 1st o. H ) ` x ) = ( F ` x ) )
42	24, 26, 41	eqfnfvd 6314	. . . . 5 |- ( ph -> ( 1st o. H ) = F )
43	42	cnveqd 5298	. . . 4 |- ( ph -> `' ( 1st o. H ) = `' F )
44	43	imaeq1d 5465	. . 3 |- ( ph -> ( `' ( 1st o. H ) " Y ) = ( `' F " Y ) )
45		fo2nd 7189	. . . . . . . . 9 |- 2nd : _V -onto-> _V
46		fofn 6117	. . . . . . . . 9 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
47	45, 46	ax-mp 5	. . . . . . . 8 |- 2nd Fn _V
48		fnco 5999	. . . . . . . 8 |- ( ( 2nd Fn _V /\ H Fn A /\ ran H C_ _V ) -> ( 2nd o. H ) Fn A )
49	47, 20, 21, 48	mp3an 1424	. . . . . . 7 |- ( 2nd o. H ) Fn A
50	49	a1i 11	. . . . . 6 |- ( ph -> ( 2nd o. H ) Fn A )
51		ffn 6045	. . . . . . 7 |- ( G : A --> C -> G Fn A )
52	5, 51	syl 17	. . . . . 6 |- ( ph -> G Fn A )
53	40	simprd 479	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( 2nd o. H ) ` x ) = ( G ` x ) )
54	50, 52, 53	eqfnfvd 6314	. . . . 5 |- ( ph -> ( 2nd o. H ) = G )
55	54	cnveqd 5298	. . . 4 |- ( ph -> `' ( 2nd o. H ) = `' G )
56	55	imaeq1d 5465	. . 3 |- ( ph -> ( `' ( 2nd o. H ) " Z ) = ( `' G " Z ) )
57	44, 56	ineq12d 3815	. 2 |- ( ph -> ( ( `' ( 1st o. H ) " Y ) i^i ( `' ( 2nd o. H ) " Z ) ) = ( ( `' F " Y ) i^i ( `' G " Z ) ) )
58	15, 57	eqtrd 2656	1 |- ( ph -> ( `' H " ( Y X. Z ) ) = ( ( `' F " Y ) i^i ( `' G " Z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   |-> cmpt 4729   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-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-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:  mbfmco2  30327