Theorem suppssrst 6439

Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)

Hypotheses

Ref	Expression
suppssr.f	|- ( ph -> F : A --> B )
suppssr.n	|- ( ph -> ( F supp Z ) C_ W )
suppssr.a	|- ( ph -> A e. V )
suppssrst.z	|- ( ph -> Z e. B )
suppssrst.st	|- ( ph -> A. u e. B A. v e. B STAB u = v )

Assertion

Ref	Expression
suppssrst	|- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Distinct variable groups:   u, B, v   u, F, v   u, X, v   v, Z

Allowed substitution hints:   ph( v, u)   A( v, u)   V( v, u)   W( v, u)   Z( u)

Proof of Theorem suppssrst

Step	Hyp	Ref	Expression
1		eldif 3210	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		df-ne 2404	. . . . . 6 |- ( ( F ` X ) =/= Z <-> -. ( F ` X ) = Z )
3		suppssr.f	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
4		suppssr.a	. . . . . . . . . . 11 |- ( ph -> A e. V )
5	3, 4	fexd 5894	. . . . . . . . . 10 |- ( ph -> F e. _V )
6		fvexg 5667	. . . . . . . . . 10 |- ( ( F e. _V /\ X e. A ) -> ( F ` X ) e. _V )
7	5, 6	sylan 283	. . . . . . . . 9 |- ( ( ph /\ X e. A ) -> ( F ` X ) e. _V )
8	7	biantrurd 305	. . . . . . . 8 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) ) )
9		eldifsn 3804	. . . . . . . 8 |- ( ( F ` X ) e. ( _V \ { Z } ) <-> ( ( F ` X ) e. _V /\ ( F ` X ) =/= Z ) )
10	8, 9	bitr4di 198	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z <-> ( F ` X ) e. ( _V \ { Z } ) ) )
11	3	ffnd 5490	. . . . . . . . . . 11 |- ( ph -> F Fn A )
12		suppssrst.z	. . . . . . . . . . 11 |- ( ph -> Z e. B )
13		elsuppfn 6421	. . . . . . . . . . 11 |- ( ( F Fn A /\ A e. V /\ Z e. B ) -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
14	11, 4, 12, 13	syl3anc 1274	. . . . . . . . . 10 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
15	10	pm5.32da 452	. . . . . . . . . 10 |- ( ph -> ( ( X e. A /\ ( F ` X ) =/= Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
16	14, 15	bitrd 188	. . . . . . . . 9 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) ) )
17		suppssr.n	. . . . . . . . . 10 |- ( ph -> ( F supp Z ) C_ W )
18	17	sseld 3227	. . . . . . . . 9 |- ( ph -> ( X e. ( F supp Z ) -> X e. W ) )
19	16, 18	sylbird 170	. . . . . . . 8 |- ( ph -> ( ( X e. A /\ ( F ` X ) e. ( _V \ { Z } ) ) -> X e. W ) )
20	19	expdimp 259	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) e. ( _V \ { Z } ) -> X e. W ) )
21	10, 20	sylbid 150	. . . . . 6 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) =/= Z -> X e. W ) )
22	2, 21	biimtrrid 153	. . . . 5 |- ( ( ph /\ X e. A ) -> ( -. ( F ` X ) = Z -> X e. W ) )
23	22	con3d 636	. . . 4 |- ( ( ph /\ X e. A ) -> ( -. X e. W -> -. -. ( F ` X ) = Z ) )
24		eqeq2 2241	. . . . . . 7 |- ( v = Z -> ( ( F ` X ) = v <-> ( F ` X ) = Z ) )
25	24	stbid 840	. . . . . 6 |- ( v = Z -> (STAB ( F ` X ) = v <-> STAB ( F ` X ) = Z ) )
26		eqeq1 2238	. . . . . . . . 9 |- ( u = ( F ` X ) -> ( u = v <-> ( F ` X ) = v ) )
27	26	stbid 840	. . . . . . . 8 |- ( u = ( F ` X ) -> (STAB u = v <-> STAB ( F ` X ) = v ) )
28	27	ralbidv 2533	. . . . . . 7 |- ( u = ( F ` X ) -> ( A. v e. B STAB u = v <-> A. v e. B STAB ( F ` X ) = v ) )
29		suppssrst.st	. . . . . . . 8 |- ( ph -> A. u e. B A. v e. B STAB u = v )
30	29	adantr 276	. . . . . . 7 |- ( ( ph /\ X e. A ) -> A. u e. B A. v e. B STAB u = v )
31	3	ffvelcdmda 5790	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( F ` X ) e. B )
32	28, 30, 31	rspcdva 2916	. . . . . 6 |- ( ( ph /\ X e. A ) -> A. v e. B STAB ( F ` X ) = v )
33	12	adantr 276	. . . . . 6 |- ( ( ph /\ X e. A ) -> Z e. B )
34	25, 32, 33	rspcdva 2916	. . . . 5 |- ( ( ph /\ X e. A ) -> STAB ( F ` X ) = Z )
35		df-stab 839	. . . . 5 |- (STAB ( F ` X ) = Z <-> ( -. -. ( F ` X ) = Z -> ( F ` X ) = Z ) )
36	34, 35	sylib 122	. . . 4 |- ( ( ph /\ X e. A ) -> ( -. -. ( F ` X ) = Z -> ( F ` X ) = Z ) )
37	23, 36	syld 45	. . 3 |- ( ( ph /\ X e. A ) -> ( -. X e. W -> ( F ` X ) = Z ) )
38	37	impr 379	. 2 |- ( ( ph /\ ( X e. A /\ -. X e. W ) ) -> ( F ` X ) = Z )
39	1, 38	sylan2b 287	1 |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 838   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   _Vcvv 2803   \ cdif 3198   C_ wss 3201   {csn 3673   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   supp csupp 6413

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-stab 839  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by: (None)