Theorem eqfnun 15705

Description: Two functions on A u. B are equal if and only if they have equal restrictions to both A and B.

Assertion

Ref	Expression
eqfnun	|- ((F Fn (A u. B) /\ G Fn (A u. B)) -> (F = G <-> ((F |` A) = (G |` A) /\ (F |` B) = (G |` B))))

Proof of Theorem eqfnun

Step	Hyp	Ref	Expression
1		reseq1 4218	. . 3 |- (F = G -> (F |` A) = (G |` A))
2		reseq1 4218	. . 3 |- (F = G -> (F |` B) = (G |` B))
3	1, 2	jca 310	. 2 |- (F = G -> ((F |` A) = (G |` A) /\ (F |` B) = (G |` B)))
4		eqfnfv2 4767	. . 3 |- ((F Fn (A u. B) /\ G Fn (A u. B)) -> (F = G <-> A.x e. (A u. B)(F` x) = (G` x)))
5		fveq1 4680	. . . . . . . . 9 |- ((F |` A) = (G |` A) -> ((F |` A)` x) = ((G |` A)` x))
6	5	adantr 425	. . . . . . . 8 |- (((F |` A) = (G |` A) /\ x e. A) -> ((F |` A)` x) = ((G |` A)` x))
7		fvres 4691	. . . . . . . . 9 |- (x e. A -> ((F |` A)` x) = (F` x))
8	7	adantl 424	. . . . . . . 8 |- (((F |` A) = (G |` A) /\ x e. A) -> ((F |` A)` x) = (F` x))
9		fvres 4691	. . . . . . . . 9 |- (x e. A -> ((G |` A)` x) = (G` x))
10	9	adantl 424	. . . . . . . 8 |- (((F |` A) = (G |` A) /\ x e. A) -> ((G |` A)` x) = (G` x))
11	6, 8, 10	3eqtr3d 1934	. . . . . . 7 |- (((F |` A) = (G |` A) /\ x e. A) -> (F` x) = (G` x))
12	11	adantlr 429	. . . . . 6 |- ((((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) /\ x e. A) -> (F` x) = (G` x))
13		fveq1 4680	. . . . . . . . 9 |- ((F |` B) = (G |` B) -> ((F |` B)` x) = ((G |` B)` x))
14	13	adantr 425	. . . . . . . 8 |- (((F |` B) = (G |` B) /\ x e. B) -> ((F |` B)` x) = ((G |` B)` x))
15		fvres 4691	. . . . . . . . 9 |- (x e. B -> ((F |` B)` x) = (F` x))
16	15	adantl 424	. . . . . . . 8 |- (((F |` B) = (G |` B) /\ x e. B) -> ((F |` B)` x) = (F` x))
17		fvres 4691	. . . . . . . . 9 |- (x e. B -> ((G |` B)` x) = (G` x))
18	17	adantl 424	. . . . . . . 8 |- (((F |` B) = (G |` B) /\ x e. B) -> ((G |` B)` x) = (G` x))
19	14, 16, 18	3eqtr3d 1934	. . . . . . 7 |- (((F |` B) = (G |` B) /\ x e. B) -> (F` x) = (G` x))
20	19	adantll 428	. . . . . 6 |- ((((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) /\ x e. B) -> (F` x) = (G` x))
21	12, 20	jaodan 471	. . . . 5 |- ((((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) /\ (x e. A \/ x e. B)) -> (F` x) = (G` x))
22		elun 2741	. . . . 5 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
23	21, 22	sylan2b 501	. . . 4 |- ((((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) /\ x e. (A u. B)) -> (F` x) = (G` x))
24	23	r19.21aiva 2176	. . 3 |- (((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) -> A.x e. (A u. B)(F` x) = (G` x))
25	4, 24	syl5bir 227	. 2 |- ((F Fn (A u. B) /\ G Fn (A u. B)) -> (((F |` A) = (G |` A) /\ (F |` B) = (G |` B)) -> F = G))
26	3, 25	impbid2 576	1 |- ((F Fn (A u. B) /\ G Fn (A u. B)) -> (F = G <-> ((F |` A) = (G |` A) /\ (F |` B) = (G |` B))))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   u. cun 2591   |` cres 3988   Fn wfn 3993  ` cfv 3998

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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