Theorem ofpreima 29465

Description: Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)

Hypotheses

Ref	Expression
ofpreima.1	|- ( ph -> F : A --> B )
ofpreima.2	|- ( ph -> G : A --> C )
ofpreima.3	|- ( ph -> A e. V )
ofpreima.4	|- ( ph -> R Fn ( B X. C ) )

Assertion

Ref	Expression
ofpreima	|- ( ph -> ( `' ( F oF R G ) " D ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )

Distinct variable groups:   A, p   D, p   F, p   G, p   R, p   ph, p

Allowed substitution hints:   B( p)   C( p)   V( p)

Proof of Theorem ofpreima

Dummy variables q s x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfmpt1 4747	. . . . . . 7 |- F/_ s ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. )
2		ofpreima.1	. . . . . . 7 |- ( ph -> F : A --> B )
3		ofpreima.2	. . . . . . 7 |- ( ph -> G : A --> C )
4		ofpreima.3	. . . . . . 7 |- ( ph -> A e. V )
5		eqidd 2623	. . . . . . 7 |- ( ph -> ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) = ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) )
6		ofpreima.4	. . . . . . . 8 |- ( ph -> R Fn ( B X. C ) )
7		fnov 6768	. . . . . . . 8 |- ( R Fn ( B X. C ) <-> R = ( x e. B , y e. C |-> ( x R y ) ) )
8	6, 7	sylib 208	. . . . . . 7 |- ( ph -> R = ( x e. B , y e. C |-> ( x R y ) ) )
9	1, 2, 3, 4, 5, 8	ofoprabco 29464	. . . . . 6 |- ( ph -> ( F oF R G ) = ( R o. ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ) )
10	9	cnveqd 5298	. . . . 5 |- ( ph -> `' ( F oF R G ) = `' ( R o. ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ) )
11		cnvco 5308	. . . . 5 |- `' ( R o. ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ) = ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) o. `' R )
12	10, 11	syl6eq 2672	. . . 4 |- ( ph -> `' ( F oF R G ) = ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) o. `' R ) )
13	12	imaeq1d 5465	. . 3 |- ( ph -> ( `' ( F oF R G ) " D ) = ( ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) o. `' R ) " D ) )
14		imaco 5640	. . 3 |- ( ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) o. `' R ) " D ) = ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) " ( `' R " D ) )
15	13, 14	syl6eq 2672	. 2 |- ( ph -> ( `' ( F oF R G ) " D ) = ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) " ( `' R " D ) ) )
16		dfima2 5468	. . 3 |- ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) " ( `' R " D ) ) = { q | E. p e. ( `' R " D ) p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q }
17		vex 3203	. . . . . . . 8 |- p e. _V
18		vex 3203	. . . . . . . 8 |- q e. _V
19	17, 18	brcnv 5305	. . . . . . 7 |- ( p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q <-> q ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) p )
20		funmpt 5926	. . . . . . . . 9 |- Fun ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. )
21		funbrfv2b 6240	. . . . . . . . 9 |- ( Fun ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) -> ( q ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) p <-> ( q e. dom ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) ) )
22	20, 21	ax-mp 5	. . . . . . . 8 |- ( q ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) p <-> ( q e. dom ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) )
23		opex 4932	. . . . . . . . . . 11 |- <. ( F ` s ) , ( G ` s ) >. e. _V
24		eqid 2622	. . . . . . . . . . 11 |- ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) = ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. )
25	23, 24	dmmpti 6023	. . . . . . . . . 10 |- dom ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) = A
26	25	eleq2i 2693	. . . . . . . . 9 |- ( q e. dom ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) <-> q e. A )
27	26	anbi1i 731	. . . . . . . 8 |- ( ( q e. dom ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) <-> ( q e. A /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) )
28	22, 27	bitri 264	. . . . . . 7 |- ( q ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) p <-> ( q e. A /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) )
29		fveq2 6191	. . . . . . . . . . 11 |- ( s = q -> ( F ` s ) = ( F ` q ) )
30		fveq2 6191	. . . . . . . . . . 11 |- ( s = q -> ( G ` s ) = ( G ` q ) )
31	29, 30	opeq12d 4410	. . . . . . . . . 10 |- ( s = q -> <. ( F ` s ) , ( G ` s ) >. = <. ( F ` q ) , ( G ` q ) >. )
32		opex 4932	. . . . . . . . . 10 |- <. ( F ` q ) , ( G ` q ) >. e. _V
33	31, 24, 32	fvmpt 6282	. . . . . . . . 9 |- ( q e. A -> ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = <. ( F ` q ) , ( G ` q ) >. )
34	33	eqeq1d 2624	. . . . . . . 8 |- ( q e. A -> ( ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p <-> <. ( F ` q ) , ( G ` q ) >. = p ) )
35	34	pm5.32i 669	. . . . . . 7 |- ( ( q e. A /\ ( ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) ` q ) = p ) <-> ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) )
36	19, 28, 35	3bitri 286	. . . . . 6 |- ( p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q <-> ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) )
37	36	rexbii 3041	. . . . 5 |- ( E. p e. ( `' R " D ) p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q <-> E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) )
38	37	abbii 2739	. . . 4 |- { q | E. p e. ( `' R " D ) p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q } = { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) }
39		nfv 1843	. . . . 5 |- F/ q ph
40		nfab1 2766	. . . . 5 |- F/_ q { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) }
41		nfcv 2764	. . . . 5 |- F/_ q U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) )
42		eliun 4524	. . . . . 6 |- ( q e. U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> E. p e. ( `' R " D ) q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
43		ffn 6045	. . . . . . . . . . . . 13 |- ( F : A --> B -> F Fn A )
44		fniniseg 6338	. . . . . . . . . . . . 13 |- ( F Fn A -> ( q e. ( `' F " { ( 1st ` p ) } ) <-> ( q e. A /\ ( F ` q ) = ( 1st ` p ) ) ) )
45	2, 43, 44	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( q e. ( `' F " { ( 1st ` p ) } ) <-> ( q e. A /\ ( F ` q ) = ( 1st ` p ) ) ) )
46		ffn 6045	. . . . . . . . . . . . 13 |- ( G : A --> C -> G Fn A )
47		fniniseg 6338	. . . . . . . . . . . . 13 |- ( G Fn A -> ( q e. ( `' G " { ( 2nd ` p ) } ) <-> ( q e. A /\ ( G ` q ) = ( 2nd ` p ) ) ) )
48	3, 46, 47	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( q e. ( `' G " { ( 2nd ` p ) } ) <-> ( q e. A /\ ( G ` q ) = ( 2nd ` p ) ) ) )
49	45, 48	anbi12d 747	. . . . . . . . . . 11 |- ( ph -> ( ( q e. ( `' F " { ( 1st ` p ) } ) /\ q e. ( `' G " { ( 2nd ` p ) } ) ) <-> ( ( q e. A /\ ( F ` q ) = ( 1st ` p ) ) /\ ( q e. A /\ ( G ` q ) = ( 2nd ` p ) ) ) ) )
50		elin 3796	. . . . . . . . . . 11 |- ( q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> ( q e. ( `' F " { ( 1st ` p ) } ) /\ q e. ( `' G " { ( 2nd ` p ) } ) ) )
51		anandi 871	. . . . . . . . . . 11 |- ( ( q e. A /\ ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) ) <-> ( ( q e. A /\ ( F ` q ) = ( 1st ` p ) ) /\ ( q e. A /\ ( G ` q ) = ( 2nd ` p ) ) ) )
52	49, 50, 51	3bitr4g 303	. . . . . . . . . 10 |- ( ph -> ( q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> ( q e. A /\ ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) ) ) )
53	52	adantr 481	. . . . . . . . 9 |- ( ( ph /\ p e. ( `' R " D ) ) -> ( q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> ( q e. A /\ ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) ) ) )
54		cnvimass 5485	. . . . . . . . . . . . . 14 |- ( `' R " D ) C_ dom R
55		fndm 5990	. . . . . . . . . . . . . . 15 |- ( R Fn ( B X. C ) -> dom R = ( B X. C ) )
56	6, 55	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> dom R = ( B X. C ) )
57	54, 56	syl5sseq 3653	. . . . . . . . . . . . 13 |- ( ph -> ( `' R " D ) C_ ( B X. C ) )
58	57	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( `' R " D ) ) -> p e. ( B X. C ) )
59		1st2nd2 7205	. . . . . . . . . . . 12 |- ( p e. ( B X. C ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
60		eqeq2 2633	. . . . . . . . . . . 12 |- ( p = <. ( 1st ` p ) , ( 2nd ` p ) >. -> ( <. ( F ` q ) , ( G ` q ) >. = p <-> <. ( F ` q ) , ( G ` q ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
61	58, 59, 60	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( `' R " D ) ) -> ( <. ( F ` q ) , ( G ` q ) >. = p <-> <. ( F ` q ) , ( G ` q ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
62		fvex 6201	. . . . . . . . . . . 12 |- ( F ` q ) e. _V
63		fvex 6201	. . . . . . . . . . . 12 |- ( G ` q ) e. _V
64	62, 63	opth 4945	. . . . . . . . . . 11 |- ( <. ( F ` q ) , ( G ` q ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. <-> ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) )
65	61, 64	syl6bb 276	. . . . . . . . . 10 |- ( ( ph /\ p e. ( `' R " D ) ) -> ( <. ( F ` q ) , ( G ` q ) >. = p <-> ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) ) )
66	65	anbi2d 740	. . . . . . . . 9 |- ( ( ph /\ p e. ( `' R " D ) ) -> ( ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) <-> ( q e. A /\ ( ( F ` q ) = ( 1st ` p ) /\ ( G ` q ) = ( 2nd ` p ) ) ) ) )
67	53, 66	bitr4d 271	. . . . . . . 8 |- ( ( ph /\ p e. ( `' R " D ) ) -> ( q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) ) )
68	67	rexbidva 3049	. . . . . . 7 |- ( ph -> ( E. p e. ( `' R " D ) q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) ) )
69		abid 2610	. . . . . . 7 |- ( q e. { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) } <-> E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) )
70	68, 69	syl6bbr 278	. . . . . 6 |- ( ph -> ( E. p e. ( `' R " D ) q e. ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) <-> q e. { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) } ) )
71	42, 70	syl5rbb 273	. . . . 5 |- ( ph -> ( q e. { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) } <-> q e. U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
72	39, 40, 41, 71	eqrd 3622	. . . 4 |- ( ph -> { q | E. p e. ( `' R " D ) ( q e. A /\ <. ( F ` q ) , ( G ` q ) >. = p ) } = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
73	38, 72	syl5eq 2668	. . 3 |- ( ph -> { q | E. p e. ( `' R " D ) p `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) q } = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
74	16, 73	syl5eq 2668	. 2 |- ( ph -> ( `' ( s e. A |-> <. ( F ` s ) , ( G ` s ) >. ) " ( `' R " D ) ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
75	15, 74	eqtrd 2656	1 |- ( ph -> ( `' ( F oF R G ) " D ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   i^i cin 3573   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   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-rep 4771  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-reu 2919  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-1st 7168  df-2nd 7169

This theorem is referenced by:  ofpreima2  29466