Theorem ofpreima2 29466

Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 29465 iterates the union over a smaller set. (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
ofpreima2	|- ( ph -> ( `' ( F oF R G ) " D ) = U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' 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 ofpreima2

Step	Hyp	Ref	Expression
1		ofpreima.1	. . . 4 |- ( ph -> F : A --> B )
2		ofpreima.2	. . . 4 |- ( ph -> G : A --> C )
3		ofpreima.3	. . . 4 |- ( ph -> A e. V )
4		ofpreima.4	. . . 4 |- ( ph -> R Fn ( B X. C ) )
5	1, 2, 3, 4	ofpreima 29465	. . 3 |- ( ph -> ( `' ( F oF R G ) " D ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
6		inundif 4046	. . . . 5 |- ( ( ( `' R " D ) i^i ( ran F X. ran G ) ) u. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) = ( `' R " D )
7		iuneq1 4534	. . . . 5 |- ( ( ( ( `' R " D ) i^i ( ran F X. ran G ) ) u. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) = ( `' R " D ) -> U_ p e. ( ( ( `' R " D ) i^i ( ran F X. ran G ) ) u. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
8	6, 7	ax-mp 5	. . . 4 |- U_ p e. ( ( ( `' R " D ) i^i ( ran F X. ran G ) ) u. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) )
9		iunxun 4605	. . . 4 |- U_ p e. ( ( ( `' R " D ) i^i ( ran F X. ran G ) ) u. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
10	8, 9	eqtr3i 2646	. . 3 |- U_ p e. ( `' R " D ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
11	5, 10	syl6eq 2672	. 2 |- ( ph -> ( `' ( F oF R G ) " D ) = ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
12		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) )
13	12	eldifbd 3587	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> -. p e. ( ran F X. ran G ) )
14		cnvimass 5485	. . . . . . . . . . . . . 14 |- ( `' R " D ) C_ dom R
15		fndm 5990	. . . . . . . . . . . . . . 15 |- ( R Fn ( B X. C ) -> dom R = ( B X. C ) )
16	4, 15	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> dom R = ( B X. C ) )
17	14, 16	syl5sseq 3653	. . . . . . . . . . . . 13 |- ( ph -> ( `' R " D ) C_ ( B X. C ) )
18	17	ssdifssd 3748	. . . . . . . . . . . 12 |- ( ph -> ( ( `' R " D ) \ ( ran F X. ran G ) ) C_ ( B X. C ) )
19	18	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> p e. ( B X. C ) )
20		1st2nd2 7205	. . . . . . . . . . 11 |- ( p e. ( B X. C ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
21		elxp6 7200	. . . . . . . . . . . 12 |- ( p e. ( ran F X. ran G ) <-> ( p = <. ( 1st ` p ) , ( 2nd ` p ) >. /\ ( ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) ) )
22	21	simplbi2 655	. . . . . . . . . . 11 |- ( p = <. ( 1st ` p ) , ( 2nd ` p ) >. -> ( ( ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) -> p e. ( ran F X. ran G ) ) )
23	19, 20, 22	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> ( ( ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) -> p e. ( ran F X. ran G ) ) )
24	13, 23	mtod 189	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> -. ( ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) )
25		ianor 509	. . . . . . . . 9 |- ( -. ( ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) <-> ( -. ( 1st ` p ) e. ran F \/ -. ( 2nd ` p ) e. ran G ) )
26	24, 25	sylib 208	. . . . . . . 8 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> ( -. ( 1st ` p ) e. ran F \/ -. ( 2nd ` p ) e. ran G ) )
27		disjsn 4246	. . . . . . . . 9 |- ( ( ran F i^i { ( 1st ` p ) } ) = (/) <-> -. ( 1st ` p ) e. ran F )
28		disjsn 4246	. . . . . . . . 9 |- ( ( ran G i^i { ( 2nd ` p ) } ) = (/) <-> -. ( 2nd ` p ) e. ran G )
29	27, 28	orbi12i 543	. . . . . . . 8 |- ( ( ( ran F i^i { ( 1st ` p ) } ) = (/) \/ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) <-> ( -. ( 1st ` p ) e. ran F \/ -. ( 2nd ` p ) e. ran G ) )
30	26, 29	sylibr 224	. . . . . . 7 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> ( ( ran F i^i { ( 1st ` p ) } ) = (/) \/ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) )
31	1	ffnd 6046	. . . . . . . . 9 |- ( ph -> F Fn A )
32		dffn3 6054	. . . . . . . . 9 |- ( F Fn A <-> F : A --> ran F )
33	31, 32	sylib 208	. . . . . . . 8 |- ( ph -> F : A --> ran F )
34	2	ffnd 6046	. . . . . . . . . 10 |- ( ph -> G Fn A )
35		dffn3 6054	. . . . . . . . . 10 |- ( G Fn A <-> G : A --> ran G )
36	34, 35	sylib 208	. . . . . . . . 9 |- ( ph -> G : A --> ran G )
37	36	adantr 481	. . . . . . . 8 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> G : A --> ran G )
38		fimacnvdisj 6083	. . . . . . . . . . 11 |- ( ( F : A --> ran F /\ ( ran F i^i { ( 1st ` p ) } ) = (/) ) -> ( `' F " { ( 1st ` p ) } ) = (/) )
39		ineq1 3807	. . . . . . . . . . . 12 |- ( ( `' F " { ( 1st ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = ( (/) i^i ( `' G " { ( 2nd ` p ) } ) ) )
40		0in 3969	. . . . . . . . . . . 12 |- ( (/) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/)
41	39, 40	syl6eq 2672	. . . . . . . . . . 11 |- ( ( `' F " { ( 1st ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
42	38, 41	syl 17	. . . . . . . . . 10 |- ( ( F : A --> ran F /\ ( ran F i^i { ( 1st ` p ) } ) = (/) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
43	42	ex 450	. . . . . . . . 9 |- ( F : A --> ran F -> ( ( ran F i^i { ( 1st ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) ) )
44		fimacnvdisj 6083	. . . . . . . . . . 11 |- ( ( G : A --> ran G /\ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) -> ( `' G " { ( 2nd ` p ) } ) = (/) )
45		ineq2 3808	. . . . . . . . . . . 12 |- ( ( `' G " { ( 2nd ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = ( ( `' F " { ( 1st ` p ) } ) i^i (/) ) )
46		in0 3968	. . . . . . . . . . . 12 |- ( ( `' F " { ( 1st ` p ) } ) i^i (/) ) = (/)
47	45, 46	syl6eq 2672	. . . . . . . . . . 11 |- ( ( `' G " { ( 2nd ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
48	44, 47	syl 17	. . . . . . . . . 10 |- ( ( G : A --> ran G /\ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
49	48	ex 450	. . . . . . . . 9 |- ( G : A --> ran G -> ( ( ran G i^i { ( 2nd ` p ) } ) = (/) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) ) )
50	43, 49	jaao 531	. . . . . . . 8 |- ( ( F : A --> ran F /\ G : A --> ran G ) -> ( ( ( ran F i^i { ( 1st ` p ) } ) = (/) \/ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) ) )
51	33, 37, 50	syl2an2r 876	. . . . . . 7 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> ( ( ( ran F i^i { ( 1st ` p ) } ) = (/) \/ ( ran G i^i { ( 2nd ` p ) } ) = (/) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) ) )
52	30, 51	mpd 15	. . . . . 6 |- ( ( ph /\ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
53	52	iuneq2dv 4542	. . . . 5 |- ( ph -> U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) (/) )
54		iun0 4576	. . . . 5 |- U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) (/) = (/)
55	53, 54	syl6eq 2672	. . . 4 |- ( ph -> U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) = (/) )
56	55	uneq2d 3767	. . 3 |- ( ph -> ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. (/) ) )
57		un0 3967	. . 3 |- ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. (/) ) = U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) )
58	56, 57	syl6eq 2672	. 2 |- ( ph -> ( U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) u. U_ p e. ( ( `' R " D ) \ ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
59	11, 58	eqtrd 2656	1 |- ( ph -> ( `' ( F oF R G ) " D ) = U_ p e. ( ( `' R " D ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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:  sibfof  30402