Theorem imacosupp 7335

Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)

Assertion

Ref	Expression
imacosupp	|- ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )

Proof of Theorem imacosupp

Step	Hyp	Ref	Expression
1		cnvco 5308	. . . . . . . 8 |- `' ( F o. G ) = ( `' G o. `' F )
2	1	imaeq1i 5463	. . . . . . 7 |- ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( ( `' G o. `' F ) " ( _V \ { Z } ) )
3		imaco 5640	. . . . . . 7 |- ( ( `' G o. `' F ) " ( _V \ { Z } ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) )
4	2, 3	eqtri 2644	. . . . . 6 |- ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) )
5	4	imaeq2i 5464	. . . . 5 |- ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) = ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) )
6		funforn 6122	. . . . . . . 8 |- ( Fun G <-> G : dom G -onto-> ran G )
7	6	biimpi 206	. . . . . . 7 |- ( Fun G -> G : dom G -onto-> ran G )
8	7	ad2antrl 764	. . . . . 6 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> G : dom G -onto-> ran G )
9		simpl 473	. . . . . . . . . . . . 13 |- ( ( F e. V /\ G e. W ) -> F e. V )
10	9	anim2i 593	. . . . . . . . . . . 12 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( Z e. _V /\ F e. V ) )
11	10	ancomd 467	. . . . . . . . . . 11 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( F e. V /\ Z e. _V ) )
12		suppimacnv 7306	. . . . . . . . . . 11 |- ( ( F e. V /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
14	13	sseq1d 3632	. . . . . . . . 9 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F supp Z ) C_ ran G <-> ( `' F " ( _V \ { Z } ) ) C_ ran G ) )
15	14	biimpd 219	. . . . . . . 8 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F supp Z ) C_ ran G -> ( `' F " ( _V \ { Z } ) ) C_ ran G ) )
16	15	adantld 483	. . . . . . 7 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( `' F " ( _V \ { Z } ) ) C_ ran G ) )
17	16	imp 445	. . . . . 6 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( `' F " ( _V \ { Z } ) ) C_ ran G )
18		foimacnv 6154	. . . . . 6 |- ( ( G : dom G -onto-> ran G /\ ( `' F " ( _V \ { Z } ) ) C_ ran G ) -> ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) ) = ( `' F " ( _V \ { Z } ) ) )
19	8, 17, 18	syl2anc 693	. . . . 5 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( `' G " ( `' F " ( _V \ { Z } ) ) ) ) = ( `' F " ( _V \ { Z } ) ) )
20	5, 19	syl5eq 2668	. . . 4 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
21		coexg 7117	. . . . . . . . 9 |- ( ( F e. V /\ G e. W ) -> ( F o. G ) e. _V )
22	21	anim2i 593	. . . . . . . 8 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( Z e. _V /\ ( F o. G ) e. _V ) )
23	22	ancomd 467	. . . . . . 7 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F o. G ) e. _V /\ Z e. _V ) )
24		suppimacnv 7306	. . . . . . 7 |- ( ( ( F o. G ) e. _V /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) )
25	23, 24	syl 17	. . . . . 6 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) )
26	25	imaeq2d 5466	. . . . 5 |- ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) )
27	26	adantr 481	. . . 4 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' ( F o. G ) " ( _V \ { Z } ) ) ) )
28	13	adantr 481	. . . 4 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
29	20, 27, 28	3eqtr4d 2666	. . 3 |- ( ( ( Z e. _V /\ ( F e. V /\ G e. W ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) )
30	29	exp31 630	. 2 |- ( Z e. _V -> ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) )
31		ima0 5481	. . . 4 |- ( G " (/) ) = (/)
32		id 22	. . . . . . 7 |- ( -. Z e. _V -> -. Z e. _V )
33	32	intnand 962	. . . . . 6 |- ( -. Z e. _V -> -. ( ( F o. G ) e. _V /\ Z e. _V ) )
34		supp0prc 7298	. . . . . 6 |- ( -. ( ( F o. G ) e. _V /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = (/) )
35	33, 34	syl 17	. . . . 5 |- ( -. Z e. _V -> ( ( F o. G ) supp Z ) = (/) )
36	35	imaeq2d 5466	. . . 4 |- ( -. Z e. _V -> ( G " ( ( F o. G ) supp Z ) ) = ( G " (/) ) )
37	32	intnand 962	. . . . 5 |- ( -. Z e. _V -> -. ( F e. _V /\ Z e. _V ) )
38		supp0prc 7298	. . . . 5 |- ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) )
39	37, 38	syl 17	. . . 4 |- ( -. Z e. _V -> ( F supp Z ) = (/) )
40	31, 36, 39	3eqtr4a 2682	. . 3 |- ( -. Z e. _V -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) )
41	40	2a1d 26	. 2 |- ( -. Z e. _V -> ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) )
42	30, 41	pm2.61i 176	1 |- ( ( F e. V /\ G e. W ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   -onto->wfo 5886  (class class class)co 6650   supp csupp 7295

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  gsumval3lem1  18306  gsumval3lem2  18307