Theorem suppssrst 6474

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 3223	. 2 |- ( X e. ( A \ W ) <-> ( X e. A /\ -. X e. W ) )
2		df-ne 2415	. . . . . 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 5921	. . . . . . . . . 10 |- ( ph -> F e. _V )
6		fvexg 5694	. . . . . . . . . 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 3825	. . . . . . . 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 5514	. . . . . . . . . . 11 |- ( ph -> F Fn A )
12		suppssrst.z	. . . . . . . . . . 11 |- ( ph -> Z e. B )
13		elsuppfn 6456	. . . . . . . . . . 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 3241	. . . . . . . . 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 2244	. . . . . . 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 2241	. . . . . . . . 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 2544	. . . . . . 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 5817	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( F ` X ) e. B )
32	28, 30, 31	rspcdva 2928	. . . . . 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 2928	. . . . 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 2205   =/= wne 2414   A.wral 2522   _Vcvv 2815   \ cdif 3211   C_ wss 3214   {csn 3694   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   supp csupp 6448

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by: (None)