Theorem eqfnfv 3782

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28.

Assertion

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

Distinct variable groups:   x,A   x,F   x,G

Proof of Theorem eqfnfv

Step	Hyp	Ref	Expression
1		eqeq12 1479	. . . . 5 |- ((dom F = A /\ dom G = B) -> (dom F = dom G <-> A = B))
2		dmeq 3300	. . . . 5 |- (F = G -> dom F = dom G)
3	1, 2	syl5bi 208	. . . 4 |- ((dom F = A /\ dom G = B) -> (F = G -> A = B))
4		fndm 3573	. . . 4 |- (F Fn A -> dom F = A)
5		fndm 3573	. . . 4 |- (G Fn B -> dom G = B)
6	3, 4, 5	syl2an 454	. . 3 |- ((F Fn A /\ G Fn B) -> (F = G -> A = B))
7		fveq1 3708	. . . . . 6 |- (F = G -> (F` x) = (G` x))
8	7	a1d 12	. . . . 5 |- (F = G -> (x e. A -> (F` x) = (G` x)))
9	8	r19.21aiv 1705	. . . 4 |- (F = G -> A.x e. A (F` x) = (G` x))
10	9	a1i 8	. . 3 |- ((F Fn A /\ G Fn B) -> (F = G -> A.x e. A (F` x) = (G` x)))
11	6, 10	jcad 598	. 2 |- ((F Fn A /\ G Fn B) -> (F = G -> (A = B /\ A.x e. A (F` x) = (G` x))))
12		visset 1804	. . . . . . . . . . . . . . . . 17 |- y e. V
13	12	fnopfvb 3739	. . . . . . . . . . . . . . . 16 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
14	13	adantlr 393	. . . . . . . . . . . . . . 15 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
15	12	fnopfvb 3739	. . . . . . . . . . . . . . . 16 |- ((G Fn A /\ x e. A) -> ((G` x) = y <-> <.x, y>. e. G))
16	15	adantll 392	. . . . . . . . . . . . . . 15 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((G` x) = y <-> <.x, y>. e. G))
17	14, 16	bibi12d 627	. . . . . . . . . . . . . 14 |- (((F Fn A /\ G Fn A) /\ x e. A) -> (((F` x) = y <-> (G` x) = y) <-> (<.x, y>. e. F <-> <.x, y>. e. G)))
18		eqeq1 1473	. . . . . . . . . . . . . 14 |- ((F` x) = (G` x) -> ((F` x) = y <-> (G` x) = y))
19	17, 18	syl5bi 208	. . . . . . . . . . . . 13 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((F` x) = (G` x) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
20	19	ex 373	. . . . . . . . . . . 12 |- ((F Fn A /\ G Fn A) -> (x e. A -> ((F` x) = (G` x) -> (<.x, y>. e. F <-> <.x, y>. e. G))))
21	20	a2d 13	. . . . . . . . . . 11 |- ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> (x e. A -> (<.x, y>. e. F <-> <.x, y>. e. G))))
22	21	com3r 35	. . . . . . . . . 10 |- (x e. A -> ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G))))
23	4	eleq2d 1533	. . . . . . . . . . . . . 14 |- (F Fn A -> (x e. dom F <-> x e. A))
24		visset 1804	. . . . . . . . . . . . . . 15 |- x e. V
25	24	opeldm 3303	. . . . . . . . . . . . . 14 |- (<.x, y>. e. F -> x e. dom F)
26	23, 25	syl5bi 208	. . . . . . . . . . . . 13 |- (F Fn A -> (<.x, y>. e. F -> x e. A))
27	26	con3d 95	. . . . . . . . . . . 12 |- (F Fn A -> (-. x e. A -> -. <.x, y>. e. F))
28		fndm 3573	. . . . . . . . . . . . . . 15 |- (G Fn A -> dom G = A)
29	28	eleq2d 1533	. . . . . . . . . . . . . 14 |- (G Fn A -> (x e. dom G <-> x e. A))
30	24	opeldm 3303	. . . . . . . . . . . . . 14 |- (<.x, y>. e. G -> x e. dom G)
31	29, 30	syl5bi 208	. . . . . . . . . . . . 13 |- (G Fn A -> (<.x, y>. e. G -> x e. A))
32	31	con3d 95	. . . . . . . . . . . 12 |- (G Fn A -> (-. x e. A -> -. <.x, y>. e. G))
33	27, 32	anim12ii 557	. . . . . . . . . . 11 |- ((F Fn A /\ G Fn A) -> (-. x e. A -> (-. <.x, y>. e. F /\ -. <.x, y>. e. G)))
34		pm5.21 675	. . . . . . . . . . . 12 |- ((-. <.x, y>. e. F /\ -. <.x, y>. e. G) -> (<.x, y>. e. F <-> <.x, y>. e. G))
35	34	a1d 12	. . . . . . . . . . 11 |- ((-. <.x, y>. e. F /\ -. <.x, y>. e. G) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
36	33, 35	syl6com 53	. . . . . . . . . 10 |- (-. x e. A -> ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G))))
37	22, 36	pm2.61i 126	. . . . . . . . 9 |- ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
38	37	19.21adv 1283	. . . . . . . 8 |- ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> A.y(<.x, y>. e. F <-> <.x, y>. e. G)))
39	38	19.20dv 1284	. . . . . . 7 |- ((F Fn A /\ G Fn A) -> (A.x(x e. A -> (F` x) = (G` x)) -> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
40		df-ral 1641	. . . . . . 7 |- (A.x e. A (F` x) = (G` x) <-> A.x(x e. A -> (F` x) = (G` x)))
41	39, 40	syl5ib 206	. . . . . 6 |- ((F Fn A /\ G Fn A) -> (A.x e. A (F` x) = (G` x) -> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
42		eqrel 3240	. . . . . . 7 |- ((Rel F /\ Rel G) -> (F = G <-> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
43		fnrel 3572	. . . . . . 7 |- (F Fn A -> Rel F)
44		fnrel 3572	. . . . . . 7 |- (G Fn A -> Rel G)
45	42, 43, 44	syl2an 454	. . . . . 6 |- ((F Fn A /\ G Fn A) -> (F = G <-> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
46	41, 45	sylibrd 204	. . . . 5 |- ((F Fn A /\ G Fn A) -> (A.x e. A (F` x) = (G` x) -> F = G))
47		fneq2 3569	. . . . . 6 |- (A = B -> (G Fn A <-> G Fn B))
48	47	biimparc 419	. . . . 5 |- ((G Fn B /\ A = B) -> G Fn A)
49	46, 48	sylan2 451	. . . 4 |- ((F Fn A /\ (G Fn B /\ A = B)) -> (A.x e. A (F` x) = (G` x) -> F = G))
50	49	exp32 377	. . 3 |- (F Fn A -> (G Fn B -> (A = B -> (A.x e. A (F` x) = (G` x) -> F = G))))
51	50	imp4b 365	. 2 |- ((F Fn A /\ G Fn B) -> ((A = B /\ A.x e. A (F` x) = (G` x)) -> F = G))
52	11, 51	impbid 514	1 |- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  A.wral 1637  <.cop 2401  dom cdm 3160  Rel wrel 3165   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  eqfnfvf 3783  fvreseq 3784  fconst2g 3830  tfr3 3911  eqfnoprval 4001  curry1 4082  df1st2 4110  df2nd2 4111  mapenlem2 4470  seq1res 6264  seq1shftid 6293  seq1seqz 6473  seq1seq0 6477  seqzeq 6487  seqzres 6492  seqzres2 6493  invfval 8201  sspn 8329  nmlno0lem 8385  phoeqi 8449  sinco 8586  cosco 8587  shftefif1olem 8661  shftefif1olemOLD 8662  dfiop2 9596  hoeqt 9603  ho01 9671  hoeq1t 9673  kbpjt 9796  nmlnop0ALT 9835  lnopco0 9844  lnopcon 9878  lnfncon 9905  hmopidmpj 9991  pjssdif2 10013  pjinvar 10029  cayleylem2 10317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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