Theorem eqfnfv 3788

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 ⋀ ∀x ∈ A (F ‘x) = (G ‘x))))

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

Proof of Theorem eqfnfv

Step	Hyp	Ref	Expression
1		eqeq12 1484	. . . . 5 ⊢ ((dom F = A ⋀ dom G = B) → (dom F = dom G ↔ A = B))
2		dmeq 3306	. . . . 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 3579	. . . 4 ⊢ (F Fn A → dom F = A)
5		fndm 3579	. . . 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 3714	. . . . . 6 ⊢ (F = G → (F ‘x) = (G ‘x))
8	7	a1d 12	. . . . 5 ⊢ (F = G → (x ∈ A → (F ‘x) = (G ‘x)))
9	8	r19.21aiv 1710	. . . 4 ⊢ (F = G → ∀x ∈ A (F ‘x) = (G ‘x))
10	9	a1i 8	. . 3 ⊢ ((F Fn A ⋀ G Fn B) → (F = G → ∀x ∈ A (F ‘x) = (G ‘x)))
11	6, 10	jcad 599	. 2 ⊢ ((F Fn A ⋀ G Fn B) → (F = G → (A = B ⋀ ∀x ∈ A (F ‘x) = (G ‘x))))
12		visset 1809	. . . . . . . . . . . . . . . . 17 ⊢ y ∈ V
13	12	fnopfvb 3745	. . . . . . . . . . . . . . . 16 ⊢ ((F Fn A ⋀ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
14	13	adantlr 393	. . . . . . . . . . . . . . 15 ⊢ (((F Fn A ⋀ G Fn A) ⋀ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
15	12	fnopfvb 3745	. . . . . . . . . . . . . . . 16 ⊢ ((G Fn A ⋀ x ∈ A) → ((G ‘x) = y ↔ ⟨x, y⟩ ∈ G))
16	15	adantll 392	. . . . . . . . . . . . . . 15 ⊢ (((F Fn A ⋀ G Fn A) ⋀ x ∈ A) → ((G ‘x) = y ↔ ⟨x, y⟩ ∈ G))
17	14, 16	bibi12d 628	. . . . . . . . . . . . . 14 ⊢ (((F Fn A ⋀ G Fn A) ⋀ x ∈ A) → (((F ‘x) = y ↔ (G ‘x) = y) ↔ (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
18		eqeq1 1478	. . . . . . . . . . . . . 14 ⊢ ((F ‘x) = (G ‘x) → ((F ‘x) = y ↔ (G ‘x) = y))
19	17, 18	syl5bi 208	. . . . . . . . . . . . 13 ⊢ (((F Fn A ⋀ G Fn A) ⋀ x ∈ A) → ((F ‘x) = (G ‘x) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
20	19	ex 373	. . . . . . . . . . . 12 ⊢ ((F Fn A ⋀ G Fn A) → (x ∈ A → ((F ‘x) = (G ‘x) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
21	20	a2d 13	. . . . . . . . . . 11 ⊢ ((F Fn A ⋀ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (x ∈ A → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
22	21	com3r 35	. . . . . . . . . 10 ⊢ (x ∈ A → ((F Fn A ⋀ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
23	4	eleq2d 1538	. . . . . . . . . . . . . 14 ⊢ (F Fn A → (x ∈ dom F ↔ x ∈ A))
24		visset 1809	. . . . . . . . . . . . . . 15 ⊢ x ∈ V
25	24	opeldm 3309	. . . . . . . . . . . . . 14 ⊢ (⟨x, y⟩ ∈ F → x ∈ dom F)
26	23, 25	syl5bi 208	. . . . . . . . . . . . 13 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
27	26	con3d 95	. . . . . . . . . . . 12 ⊢ (F Fn A → (¬ x ∈ A → ¬ ⟨x, y⟩ ∈ F))
28		fndm 3579	. . . . . . . . . . . . . . 15 ⊢ (G Fn A → dom G = A)
29	28	eleq2d 1538	. . . . . . . . . . . . . 14 ⊢ (G Fn A → (x ∈ dom G ↔ x ∈ A))
30	24	opeldm 3309	. . . . . . . . . . . . . 14 ⊢ (⟨x, y⟩ ∈ G → x ∈ dom G)
31	29, 30	syl5bi 208	. . . . . . . . . . . . 13 ⊢ (G Fn A → (⟨x, y⟩ ∈ G → x ∈ A))
32	31	con3d 95	. . . . . . . . . . . 12 ⊢ (G Fn A → (¬ x ∈ A → ¬ ⟨x, y⟩ ∈ G))
33	27, 32	anim12ii 558	. . . . . . . . . . 11 ⊢ ((F Fn A ⋀ G Fn A) → (¬ x ∈ A → (¬ ⟨x, y⟩ ∈ F ⋀ ¬ ⟨x, y⟩ ∈ G)))
34		pm5.21 676	. . . . . . . . . . . 12 ⊢ ((¬ ⟨x, y⟩ ∈ F ⋀ ¬ ⟨x, y⟩ ∈ G) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))
35	34	a1d 12	. . . . . . . . . . 11 ⊢ ((¬ ⟨x, y⟩ ∈ F ⋀ ¬ ⟨x, y⟩ ∈ G) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
36	33, 35	syl6com 53	. . . . . . . . . 10 ⊢ (¬ x ∈ A → ((F Fn A ⋀ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
37	22, 36	pm2.61i 126	. . . . . . . . 9 ⊢ ((F Fn A ⋀ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
38	37	19.21adv 1286	. . . . . . . 8 ⊢ ((F Fn A ⋀ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → ∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
39	38	19.20dv 1287	. . . . . . 7 ⊢ ((F Fn A ⋀ G Fn A) → (∀x(x ∈ A → (F ‘x) = (G ‘x)) → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
40		df-ral 1646	. . . . . . 7 ⊢ (∀x ∈ A (F ‘x) = (G ‘x) ↔ ∀x(x ∈ A → (F ‘x) = (G ‘x)))
41	39, 40	syl5ib 206	. . . . . 6 ⊢ ((F Fn A ⋀ G Fn A) → (∀x ∈ A (F ‘x) = (G ‘x) → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
42		eqrel 3245	. . . . . . 7 ⊢ ((Rel F ⋀ Rel G) → (F = G ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
43		fnrel 3578	. . . . . . 7 ⊢ (F Fn A → Rel F)
44		fnrel 3578	. . . . . . 7 ⊢ (G Fn A → Rel G)
45	42, 43, 44	syl2an 454	. . . . . 6 ⊢ ((F Fn A ⋀ G Fn A) → (F = G ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
46	41, 45	sylibrd 204	. . . . 5 ⊢ ((F Fn A ⋀ G Fn A) → (∀x ∈ A (F ‘x) = (G ‘x) → F = G))
47		fneq2 3575	. . . . . 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)) → (∀x ∈ A (F ‘x) = (G ‘x) → F = G))
50	49	exp32 377	. . 3 ⊢ (F Fn A → (G Fn B → (A = B → (∀x ∈ A (F ‘x) = (G ‘x) → F = G))))
51	50	imp4b 365	. 2 ⊢ ((F Fn A ⋀ G Fn B) → ((A = B ⋀ ∀x ∈ A (F ‘x) = (G ‘x)) → F = G))
52	11, 51	impbid 515	1 ⊢ ((F Fn A ⋀ G Fn B) → (F = G ↔ (A = B ⋀ ∀x ∈ A (F ‘x) = (G ‘x))))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∀wral 1642  ⟨cop 2407  dom cdm 3165  Rel wrel 3170   Fn wfn 3172   ‘cfv 3177

This theorem is referenced by:  eqfnfvf 3789  fvreseq 3790  fconst2g 3836  tfr3 3917  eqfnoprval 4007  curry1 4088  df1st2 4116  df2nd2 4117  mapenlem2 4476  seq1res 6272  seq1shftid 6301  seq1seqz 6481  seq1seq0 6485  seqzeq 6495  seqzres 6500  seqzres2 6501  invfval 8213  sspn 8342  nmlno0lem 8398  phoeqi 8462  sinco 8605  cosco 8606  shftefif1olem 8680  dfiop2 9619  hoeqt 9626  ho01 9694  hoeq1t 9696  kbpjt 9819  nmlnop0ALT 9858  lnopco0 9867  lnopcon 9901  lnfncon 9928  hmopidmpj 10018  pjssdif2 10040  pjinvar 10057  cayleylem2 10344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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