Theorem ressuppss 7314

Description: The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)

Assertion

Ref	Expression
ressuppss	|- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )

Proof of Theorem ressuppss

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . . . . 9 |- ( b e. ( B i^i dom F ) <-> ( b e. B /\ b e. dom F ) )
2	1	simprbi 480	. . . . . . . 8 |- ( b e. ( B i^i dom F ) -> b e. dom F )
3		dmres 5419	. . . . . . . 8 |- dom ( F |` B ) = ( B i^i dom F )
4	2, 3	eleq2s 2719	. . . . . . 7 |- ( b e. dom ( F |` B ) -> b e. dom F )
5	4	ad2antrl 764	. . . . . 6 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> b e. dom F )
6		snssi 4339	. . . . . . . . . . . 12 |- ( b e. B -> { b } C_ B )
7		resima2 5432	. . . . . . . . . . . 12 |- ( { b } C_ B -> ( ( F |` B ) " { b } ) = ( F " { b } ) )
8	6, 7	syl 17	. . . . . . . . . . 11 |- ( b e. B -> ( ( F |` B ) " { b } ) = ( F " { b } ) )
9	8	neeq1d 2853	. . . . . . . . . 10 |- ( b e. B -> ( ( ( F |` B ) " { b } ) =/= { Z } <-> ( F " { b } ) =/= { Z } ) )
10	9	biimpd 219	. . . . . . . . 9 |- ( b e. B -> ( ( ( F |` B ) " { b } ) =/= { Z } -> ( F " { b } ) =/= { Z } ) )
11	10	adantld 483	. . . . . . . 8 |- ( b e. B -> ( ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) -> ( F " { b } ) =/= { Z } ) )
12	11	adantld 483	. . . . . . 7 |- ( b e. B -> ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( F " { b } ) =/= { Z } ) )
13		pm2.24 121	. . . . . . . . . . . 12 |- ( b e. B -> ( -. b e. B -> ( F " { b } ) =/= { Z } ) )
14	13	adantr 481	. . . . . . . . . . 11 |- ( ( b e. B /\ b e. dom F ) -> ( -. b e. B -> ( F " { b } ) =/= { Z } ) )
15	1, 14	sylbi 207	. . . . . . . . . 10 |- ( b e. ( B i^i dom F ) -> ( -. b e. B -> ( F " { b } ) =/= { Z } ) )
16	15, 3	eleq2s 2719	. . . . . . . . 9 |- ( b e. dom ( F |` B ) -> ( -. b e. B -> ( F " { b } ) =/= { Z } ) )
17	16	ad2antrl 764	. . . . . . . 8 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( -. b e. B -> ( F " { b } ) =/= { Z } ) )
18	17	com12 32	. . . . . . 7 |- ( -. b e. B -> ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( F " { b } ) =/= { Z } ) )
19	12, 18	pm2.61i 176	. . . . . 6 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( F " { b } ) =/= { Z } )
20	5, 19	jca 554	. . . . 5 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( b e. dom F /\ ( F " { b } ) =/= { Z } ) )
21	20	ex 450	. . . 4 |- ( ( F e. V /\ Z e. W ) -> ( ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) -> ( b e. dom F /\ ( F " { b } ) =/= { Z } ) ) )
22	21	ss2abdv 3675	. . 3 |- ( ( F e. V /\ Z e. W ) -> { b | ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) } C_ { b | ( b e. dom F /\ ( F " { b } ) =/= { Z } ) } )
23		df-rab 2921	. . 3 |- { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } = { b | ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) }
24		df-rab 2921	. . 3 |- { b e. dom F | ( F " { b } ) =/= { Z } } = { b | ( b e. dom F /\ ( F " { b } ) =/= { Z } ) }
25	22, 23, 24	3sstr4g 3646	. 2 |- ( ( F e. V /\ Z e. W ) -> { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } C_ { b e. dom F | ( F " { b } ) =/= { Z } } )
26		resexg 5442	. . 3 |- ( F e. V -> ( F |` B ) e. _V )
27		suppval 7297	. . 3 |- ( ( ( F |` B ) e. _V /\ Z e. W ) -> ( ( F |` B ) supp Z ) = { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } )
28	26, 27	sylan 488	. 2 |- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) = { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } )
29		suppval 7297	. 2 |- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) = { b e. dom F | ( F " { b } ) =/= { Z } } )
30	25, 28, 29	3sstr4d 3648	1 |- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   dom cdm 5114   |` cres 5116   "cima 5117  (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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  fsuppres  8300  gsumzres  18310  gsumzadd  18322  gsum2dlem2  18370  tsmsres  21947