Theorem fvreseq 2882

Description: Equality of restricted functions is determined by their values.

Assertion

Ref	Expression
fvreseq	|- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))

Distinct variable group(s):   x,B   x,F   x,G

Proof of Theorem fvreseq

Step	Hyp	Ref	Expression
1		fnssres 2734	. . . 4 |- ((F Fn A /\ B (_ A) -> (F |` B) Fn B)
2		fnssres 2734	. . . 4 |- ((G Fn A /\ B (_ A) -> (G |` B) Fn B)
3	1, 2	anim12i 268	. . 3 |- (((F Fn A /\ B (_ A) /\ (G Fn A /\ B (_ A)) -> ((F |` B) Fn B /\ (G |` B) Fn B))
4	3	anandirs 395	. 2 |- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) Fn B /\ (G |` B) Fn B))
5		cleqfv 2880	. . 3 |- (((F |` B) Fn B /\ (G |` B) Fn B) -> ((F |` B) = (G |` B) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x))))
6		fvres 2840	. . . . . 6 |- (x e. B -> ((F |` B)` x) = (F` x))
7		fvres 2840	. . . . . 6 |- (x e. B -> ((G |` B)` x) = (G` x))
8	6, 7	cleq12d 1115	. . . . 5 |- (x e. B -> (((F |` B)` x) = ((G |` B)` x) <-> (F` x) = (G` x)))
9	8	birala 1228	. . . 4 |- (A.x e. B ((F |` B)` x) = ((G |` B)` x) <-> A.x e. B (F` x) = (G` x))
10		cleqid 1102	. . . . 5 |- B = B
11	10	biantrur 544	. . . 4 |- (A.x e. B ((F |` B)` x) = ((G |` B)` x) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x)))
12	9, 11	bitr3 153	. . 3 |- (A.x e. B (F` x) = (G` x) <-> (B = B /\ A.x e. B ((F |` B)` x) = ((G |` B)` x)))
13	5, 12	syl6bbr 416	. 2 |- (((F |` B) Fn B /\ (G |` B) Fn B) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))
14	4, 13	syl 12	1 |- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487   |` cres 2412   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  tfrlem1 2949  tfr3 2964

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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