Theorem fvdifsuppst 6422

Description: Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)

Hypotheses

Ref	Expression
fvdifsuppst.1	|- ( ph -> F : A --> B )
fvdifsupp.2	|- ( ph -> A e. V )
fvdifsuppst.st	|- ( ph -> A. x e. B A. y e. B STAB x = y )
fvdifsuppst.3	|- ( ph -> Z e. B )
fvdifsupp.4	|- ( ph -> X e. ( A \ ( F supp Z ) ) )

Assertion

Ref	Expression
fvdifsuppst	|- ( ph -> ( F ` X ) = Z )

Distinct variable groups:   x, B, y   x, F, y   x, X, y   x, Z, y

Allowed substitution hints:   ph( x, y)   A( x, y)   V( x, y)

Proof of Theorem fvdifsuppst

Step	Hyp	Ref	Expression
1		fvdifsupp.4	. . . 4 |- ( ph -> X e. ( A \ ( F supp Z ) ) )
2	1	eldifbd 3213	. . 3 |- ( ph -> -. X e. ( F supp Z ) )
3		df-ne 2404	. . . 4 |- ( ( F ` X ) =/= Z <-> -. ( F ` X ) = Z )
4	1	eldifad 3212	. . . . . 6 |- ( ph -> X e. A )
5		fvdifsuppst.1	. . . . . . . 8 |- ( ph -> F : A --> B )
6	5	ffnd 5490	. . . . . . 7 |- ( ph -> F Fn A )
7		fvdifsupp.2	. . . . . . 7 |- ( ph -> A e. V )
8		fvdifsuppst.3	. . . . . . 7 |- ( ph -> Z e. B )
9		elsuppfn 6421	. . . . . . 7 |- ( ( F Fn A /\ A e. V /\ Z e. B ) -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
10	6, 7, 8, 9	syl3anc 1274	. . . . . 6 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
11	4, 10	mpbirand 441	. . . . 5 |- ( ph -> ( X e. ( F supp Z ) <-> ( F ` X ) =/= Z ) )
12	11	biimprd 158	. . . 4 |- ( ph -> ( ( F ` X ) =/= Z -> X e. ( F supp Z ) ) )
13	3, 12	biimtrrid 153	. . 3 |- ( ph -> ( -. ( F ` X ) = Z -> X e. ( F supp Z ) ) )
14	2, 13	mtod 669	. 2 |- ( ph -> -. -. ( F ` X ) = Z )
15		fvdifsuppst.st	. . . 4 |- ( ph -> A. x e. B A. y e. B STAB x = y )
16	5, 4	ffvelcdmd 5791	. . . . 5 |- ( ph -> ( F ` X ) e. B )
17		eqeq12 2244	. . . . . . 7 |- ( ( x = ( F ` X ) /\ y = Z ) -> ( x = y <-> ( F ` X ) = Z ) )
18	17	stbid 840	. . . . . 6 |- ( ( x = ( F ` X ) /\ y = Z ) -> (STAB x = y <-> STAB ( F ` X ) = Z ) )
19	18	rspc2gv 2923	. . . . 5 |- ( ( ( F ` X ) e. B /\ Z e. B ) -> ( A. x e. B A. y e. B STAB x = y -> STAB ( F ` X ) = Z ) )
20	16, 8, 19	syl2anc 411	. . . 4 |- ( ph -> ( A. x e. B A. y e. B STAB x = y -> STAB ( F ` X ) = Z ) )
21	15, 20	mpd 13	. . 3 |- ( ph -> STAB ( F ` X ) = Z )
22		df-stab 839	. . 3 |- (STAB ( F ` X ) = Z <-> ( -. -. ( F ` X ) = Z -> ( F ` X ) = Z ) )
23	21, 22	sylib 122	. 2 |- ( ph -> ( -. -. ( F ` X ) = Z -> ( F ` X ) = Z ) )
24	14, 23	mpd 13	1 |- ( ph -> ( 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   \ cdif 3198   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)