Theorem cleqfv 2880

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

Assertion

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

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

Proof of Theorem cleqfv

Step	Hyp	Ref	Expression
1		cleq12 1113	. . . . 5 |- ((dom F = A /\ dom G = B) -> (dom F = dom G <-> A = B))
2		dmeq 2531	. . . . 5 |- (F = G -> dom F = dom G)
3	1, 2	syl5bi 183	. . . 4 |- ((dom F = A /\ dom G = B) -> (F = G -> A = B))
4		fndm 2723	. . . 4 |- (F Fn A -> dom F = A)
5		fndm 2723	. . . 4 |- (G Fn B -> dom G = B)
6	3, 4, 5	syl2an 349	. . 3 |- ((F Fn A /\ G Fn B) -> (F = G -> A = B))
7		fveq1 2831	. . . . . 6 |- (F = G -> (F` x) = (G` x))
8	7	a1d 14	. . . . 5 |- (F = G -> (x e. A -> (F` x) = (G` x)))
9	8	r19.21aiv 1259	. . . 4 |- (F = G -> A.x e. A (F` x) = (G` x))
10	9	a1i 7	. . 3 |- ((F Fn A /\ G Fn B) -> (F = G -> A.x e. A (F` x) = (G` x)))
11	6, 10	jcad 455	. 2 |- ((F Fn A /\ G Fn B) -> (F = G -> (A = B /\ A.x e. A (F` x) = (G` x))))
12		visset 1350	. . . . . . . . . . . . . . . . 17 |- y e. V
13	12	fnfvop 2856	. . . . . . . . . . . . . . . 16 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
14	13	adantlr 310	. . . . . . . . . . . . . . 15 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
15	12	fnfvop 2856	. . . . . . . . . . . . . . . 16 |- ((G Fn A /\ x e. A) -> ((G` x) = y <-> <.x, y>. e. G))
16	15	adantll 309	. . . . . . . . . . . . . . 15 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((G` x) = y <-> <.x, y>. e. G))
17	14, 16	bibi12d 477	. . . . . . . . . . . . . 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		cleq1 1107	. . . . . . . . . . . . . 14 |- ((F` x) = (G` x) -> ((F` x) = y <-> (G` x) = y))
19	17, 18	syl5bi 183	. . . . . . . . . . . . 13 |- (((F Fn A /\ G Fn A) /\ x e. A) -> ((F` x) = (G` x) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
20	19	exp 291	. . . . . . . . . . . 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 15	. . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . 16 |- (F Fn A -> (x e. dom F <-> x e. A))
24		visset 1350	. . . . . . . . . . . . . . . . 17 |- x e. V
25	24	opeldm 2534	. . . . . . . . . . . . . . . 16 |- (<.x, y>. e. F -> x e. dom F)
26	23, 25	syl5bi 183	. . . . . . . . . . . . . . 15 |- (F Fn A -> (<.x, y>. e. F -> x e. A))
27	26	con3d 87	. . . . . . . . . . . . . 14 |- (F Fn A -> (-. x e. A -> -. <.x, y>. e. F))
28	27	adantr 306	. . . . . . . . . . . . 13 |- ((F Fn A /\ G Fn A) -> (-. x e. A -> -. <.x, y>. e. F))
29		fndm 2723	. . . . . . . . . . . . . . . . 17 |- (G Fn A -> dom G = A)
30	29	eleq2d 1156	. . . . . . . . . . . . . . . 16 |- (G Fn A -> (x e. dom G <-> x e. A))
31	24	opeldm 2534	. . . . . . . . . . . . . . . 16 |- (<.x, y>. e. G -> x e. dom G)
32	30, 31	syl5bi 183	. . . . . . . . . . . . . . 15 |- (G Fn A -> (<.x, y>. e. G -> x e. A))
33	32	con3d 87	. . . . . . . . . . . . . 14 |- (G Fn A -> (-. x e. A -> -. <.x, y>. e. G))
34	33	adantl 305	. . . . . . . . . . . . 13 |- ((F Fn A /\ G Fn A) -> (-. x e. A -> -. <.x, y>. e. G))
35	28, 34	jcad 455	. . . . . . . . . . . 12 |- ((F Fn A /\ G Fn A) -> (-. x e. A -> (-. <.x, y>. e. F /\ -. <.x, y>. e. G)))
36		pm5.21 502	. . . . . . . . . . . . 13 |- ((-. <.x, y>. e. F /\ -. <.x, y>. e. G) -> (<.x, y>. e. F <-> <.x, y>. e. G))
37	36	a1d 14	. . . . . . . . . . . 12 |- ((-. <.x, y>. e. F /\ -. <.x, y>. e. G) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
38	35, 37	syl6 23	. . . . . . . . . . 11 |- ((F Fn A /\ G Fn A) -> (-. x e. A -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G))))
39	38	com12 13	. . . . . . . . . 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))))
40	22, 39	pm2.61i 110	. . . . . . . . 9 |- ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> (<.x, y>. e. F <-> <.x, y>. e. G)))
41	40	19.21adv 945	. . . . . . . 8 |- ((F Fn A /\ G Fn A) -> ((x e. A -> (F` x) = (G` x)) -> A.y(<.x, y>. e. F <-> <.x, y>. e. G)))
42	41	19.20dv 946	. . . . . . 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)))
43		df-ral 1205	. . . . . . 7 |- (A.x e. A (F` x) = (G` x) <-> A.x(x e. A -> (F` x) = (G` x)))
44	42, 43	syl5ib 181	. . . . . 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)))
45		cleqrel 2483	. . . . . . 7 |- ((Rel F /\ Rel G) -> (F = G <-> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
46		fnrel 2722	. . . . . . 7 |- (F Fn A -> Rel F)
47		fnrel 2722	. . . . . . 7 |- (G Fn A -> Rel G)
48	45, 46, 47	syl2an 349	. . . . . 6 |- ((F Fn A /\ G Fn A) -> (F = G <-> A.xA.y(<.x, y>. e. F <-> <.x, y>. e. G)))
49	44, 48	sylibrd 179	. . . . 5 |- ((F Fn A /\ G Fn A) -> (A.x e. A (F` x) = (G` x) -> F = G))
50		fneq2 2719	. . . . . 6 |- (A = B -> (G Fn A <-> G Fn B))
51	50	biimparc 327	. . . . 5 |- ((G Fn B /\ A = B) -> G Fn A)
52	49, 51	sylan2 346	. . . 4 |- ((F Fn A /\ (G Fn B /\ A = B)) -> (A.x e. A (F` x) = (G` x) -> F = G))
53	52	exp32 294	. . 3 |- (F Fn A -> (G Fn B -> (A = B -> (A.x e. A (F` x) = (G` x) -> F = G))))
54	53	imp4b 283	. 2 |- ((F Fn A /\ G Fn B) -> ((A = B /\ A.x e. A (F` x) = (G` x)) -> F = G))
55	11, 54	impbid 397	1 |- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  A.wral 1201  <.cop 1810  dom cdm 2410  Rel wrel 2415   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  cleqfvf 2881  fvreseq 2882  fconst2 2902  tfr3 2964  df1st2 3098  mapenlem2 3385  hoeq 5595  ho1 5613

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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