Theorem suppcofn 6500

Description: The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) (Revised by SN, 15-Sep-2023.)

Assertion

Ref	Expression
suppcofn	|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )

Proof of Theorem suppcofn

Dummy variables x f i z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6470	. . . . 5 |- supp = ( f e. _V , z e. _V |-> { i e. dom f | ( f " { i } ) =/= { z } } )
2	1	elmpocl2 6280	. . . 4 |- ( x e. ( ( F o. G ) supp Z ) -> Z e. _V )
3	2	a1i 9	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( x e. ( ( F o. G ) supp Z ) -> Z e. _V ) )
4		simprr 537	. . . . . . . 8 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> Fun G )
5	4	funfnd 5406	. . . . . . 7 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> G Fn dom G )
6		elpreima 5822	. . . . . . 7 |- ( G Fn dom G -> ( x e. ( `' G " ( F supp Z ) ) <-> ( x e. dom G /\ ( G ` x ) e. ( F supp Z ) ) ) )
7	5, 6	syl 14	. . . . . 6 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( x e. ( `' G " ( F supp Z ) ) <-> ( x e. dom G /\ ( G ` x ) e. ( F supp Z ) ) ) )
8	7	simplbda 384	. . . . 5 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ x e. ( `' G " ( F supp Z ) ) ) -> ( G ` x ) e. ( F supp Z ) )
9	1	elmpocl2 6280	. . . . 5 |- ( ( G ` x ) e. ( F supp Z ) -> Z e. _V )
10	8, 9	syl 14	. . . 4 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ x e. ( `' G " ( F supp Z ) ) ) -> Z e. _V )
11	10	ex 115	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( x e. ( `' G " ( F supp Z ) ) -> Z e. _V ) )
12		funco 5415	. . . . . . . . . 10 |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
13	12	adantl 277	. . . . . . . . 9 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> Fun ( F o. G ) )
14	13	funfnd 5406	. . . . . . . 8 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( F o. G ) Fn dom ( F o. G ) )
15	14	adantr 276	. . . . . . 7 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( F o. G ) Fn dom ( F o. G ) )
16		coexg 5330	. . . . . . . 8 |- ( ( F e. V /\ G e. W ) -> ( F o. G ) e. _V )
17	16	ad2antrr 492	. . . . . . 7 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( F o. G ) e. _V )
18		simpr 110	. . . . . . 7 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> Z e. _V )
19		suppimacnvfn 6480	. . . . . . 7 |- ( ( ( F o. G ) Fn dom ( F o. G ) /\ ( F o. G ) e. _V /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) )
20	15, 17, 18, 19	syl3anc 1278	. . . . . 6 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = ( `' ( F o. G ) " ( _V \ { Z } ) ) )
21		cnvco 4963	. . . . . . . 8 |- `' ( F o. G ) = ( `' G o. `' F )
22	21	imaeq1i 5121	. . . . . . 7 |- ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( ( `' G o. `' F ) " ( _V \ { Z } ) )
23	22	a1i 9	. . . . . 6 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( `' ( F o. G ) " ( _V \ { Z } ) ) = ( ( `' G o. `' F ) " ( _V \ { Z } ) ) )
24		imaco 5291	. . . . . . 7 |- ( ( `' G o. `' F ) " ( _V \ { Z } ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) )
25		simprl 535	. . . . . . . . . . 11 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> Fun F )
26	25	funfnd 5406	. . . . . . . . . 10 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> F Fn dom F )
27	26	adantr 276	. . . . . . . . 9 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> F Fn dom F )
28		simplll 539	. . . . . . . . 9 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> F e. V )
29		suppimacnvfn 6480	. . . . . . . . 9 |- ( ( F Fn dom F /\ F e. V /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
30	27, 28, 18, 29	syl3anc 1278	. . . . . . . 8 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
31	30	imaeq2d 5124	. . . . . . 7 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( `' G " ( F supp Z ) ) = ( `' G " ( `' F " ( _V \ { Z } ) ) ) )
32	24, 31	eqtr4id 2290	. . . . . 6 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( ( `' G o. `' F ) " ( _V \ { Z } ) ) = ( `' G " ( F supp Z ) ) )
33	20, 23, 32	3eqtrd 2275	. . . . 5 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
34	33	eleq2d 2308	. . . 4 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ Z e. _V ) -> ( x e. ( ( F o. G ) supp Z ) <-> x e. ( `' G " ( F supp Z ) ) ) )
35	34	ex 115	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( Z e. _V -> ( x e. ( ( F o. G ) supp Z ) <-> x e. ( `' G " ( F supp Z ) ) ) ) )
36	3, 11, 35	pm5.21ndd 717	. 2 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( x e. ( ( F o. G ) supp Z ) <-> x e. ( `' G " ( F supp Z ) ) ) )
37	36	eqrdv 2236	1 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   \ cdif 3217   {csn 3708   `'ccnv 4771   dom cdm 4772   "cima 4775   o. ccom 4776   Fun wfun 5369   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   supp csupp 6469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  supp0cosupp0fn  6501  imacosuppfn  6502  fsuppcorn  7295