Theorem fvdifsuppst 6474

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 3232	. . 3 |- ( ph -> -. X e. ( F supp Z ) )
3		df-ne 2421	. . . 4 |- ( ( F ` X ) =/= Z <-> -. ( F ` X ) = Z )
4	1	eldifad 3231	. . . . . 6 |- ( ph -> X e. A )
5		fvdifsuppst.1	. . . . . . . 8 |- ( ph -> F : A --> B )
6	5	ffnd 5529	. . . . . . 7 |- ( ph -> F Fn A )
7		fvdifsupp.2	. . . . . . 7 |- ( ph -> A e. V )
8		fvdifsuppst.3	. . . . . . 7 |- ( ph -> Z e. B )
9		elsuppfn 6473	. . . . . . 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 1278	. . . . . 6 |- ( ph -> ( X e. ( F supp Z ) <-> ( X e. A /\ ( F ` X ) =/= Z ) ) )
11	4, 10	mpbirand 445	. . . . 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 673	. 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 5835	. . . . 5 |- ( ph -> ( F ` X ) e. B )
17		eqeq12 2251	. . . . . . 7 |- ( ( x = ( F ` X ) /\ y = Z ) -> ( x = y <-> ( F ` X ) = Z ) )
18	17	stbid 844	. . . . . 6 |- ( ( x = ( F ` X ) /\ y = Z ) -> (STAB x = y <-> STAB ( F ` X ) = Z ) )
19	18	rspc2gv 2942	. . . . 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 415	. . . 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 843	. . 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 842   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   \ cdif 3217   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   supp csupp 6465

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-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-stab 843  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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-supp 6466

This theorem is referenced by: (None)