Theorem eqfnfv 5393

Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
eqfnfv	⊢ ((F Fn A ∧ G Fn A) → (F = G ↔ ∀x ∈ A (F ‘x) = (G ‘x)))

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

Proof of Theorem eqfnfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5328	. . 3 ⊢ (F = G → (F ‘x) = (G ‘x))
2	1	ralrimivw 2699	. 2 ⊢ (F = G → ∀x ∈ A (F ‘x) = (G ‘x))
3		pm2.27 35	. . . . . . . . 9 ⊢ (x ∈ A → ((x ∈ A → (F ‘x) = (G ‘x)) → (F ‘x) = (G ‘x)))
4	3	adantl 452	. . . . . . . 8 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (F ‘x) = (G ‘x)))
5		eqeq1 2359	. . . . . . . . 9 ⊢ ((F ‘x) = (G ‘x) → ((F ‘x) = y ↔ (G ‘x) = y))
6		fnopfvb 5360	. . . . . . . . . . 11 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
7	6	adantlr 695	. . . . . . . . . 10 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
8		fnopfvb 5360	. . . . . . . . . . 11 ⊢ ((G Fn A ∧ x ∈ A) → ((G ‘x) = y ↔ ⟨x, y⟩ ∈ G))
9	8	adantll 694	. . . . . . . . . 10 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → ((G ‘x) = y ↔ ⟨x, y⟩ ∈ G))
10	7, 9	bibi12d 312	. . . . . . . . 9 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → (((F ‘x) = y ↔ (G ‘x) = y) ↔ (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
11	5, 10	syl5ib 210	. . . . . . . 8 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → ((F ‘x) = (G ‘x) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
12	4, 11	syld 40	. . . . . . 7 ⊢ (((F Fn A ∧ G Fn A) ∧ x ∈ A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
13	12	expcom 424	. . . . . 6 ⊢ (x ∈ A → ((F Fn A ∧ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
14		opeldm 4911	. . . . . . . . . . . . 13 ⊢ (⟨x, y⟩ ∈ F → x ∈ dom F)
15		fndm 5183	. . . . . . . . . . . . . 14 ⊢ (F Fn A → dom F = A)
16	15	eleq2d 2420	. . . . . . . . . . . . 13 ⊢ (F Fn A → (x ∈ dom F ↔ x ∈ A))
17	14, 16	syl5ib 210	. . . . . . . . . . . 12 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
18	17	adantr 451	. . . . . . . . . . 11 ⊢ ((F Fn A ∧ G Fn A) → (⟨x, y⟩ ∈ F → x ∈ A))
19	18	con3d 125	. . . . . . . . . 10 ⊢ ((F Fn A ∧ G Fn A) → (¬ x ∈ A → ¬ ⟨x, y⟩ ∈ F))
20	19	impcom 419	. . . . . . . . 9 ⊢ ((¬ x ∈ A ∧ (F Fn A ∧ G Fn A)) → ¬ ⟨x, y⟩ ∈ F)
21		opeldm 4911	. . . . . . . . . . . . 13 ⊢ (⟨x, y⟩ ∈ G → x ∈ dom G)
22		fndm 5183	. . . . . . . . . . . . . 14 ⊢ (G Fn A → dom G = A)
23	22	eleq2d 2420	. . . . . . . . . . . . 13 ⊢ (G Fn A → (x ∈ dom G ↔ x ∈ A))
24	21, 23	syl5ib 210	. . . . . . . . . . . 12 ⊢ (G Fn A → (⟨x, y⟩ ∈ G → x ∈ A))
25	24	adantl 452	. . . . . . . . . . 11 ⊢ ((F Fn A ∧ G Fn A) → (⟨x, y⟩ ∈ G → x ∈ A))
26	25	con3d 125	. . . . . . . . . 10 ⊢ ((F Fn A ∧ G Fn A) → (¬ x ∈ A → ¬ ⟨x, y⟩ ∈ G))
27	26	impcom 419	. . . . . . . . 9 ⊢ ((¬ x ∈ A ∧ (F Fn A ∧ G Fn A)) → ¬ ⟨x, y⟩ ∈ G)
28	20, 27	2falsed 340	. . . . . . . 8 ⊢ ((¬ x ∈ A ∧ (F Fn A ∧ G Fn A)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))
29	28	ex 423	. . . . . . 7 ⊢ (¬ x ∈ A → ((F Fn A ∧ G Fn A) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
30	29	a1dd 42	. . . . . 6 ⊢ (¬ x ∈ A → ((F Fn A ∧ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))))
31	13, 30	pm2.61i 156	. . . . 5 ⊢ ((F Fn A ∧ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
32	31	alrimdv 1633	. . . 4 ⊢ ((F Fn A ∧ G Fn A) → ((x ∈ A → (F ‘x) = (G ‘x)) → ∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
33	32	alimdv 1621	. . 3 ⊢ ((F Fn A ∧ G Fn A) → (∀x(x ∈ A → (F ‘x) = (G ‘x)) → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G)))
34		df-ral 2620	. . 3 ⊢ (∀x ∈ A (F ‘x) = (G ‘x) ↔ ∀x(x ∈ A → (F ‘x) = (G ‘x)))
35		eqrel 4846	. . 3 ⊢ (F = G ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ G))
36	33, 34, 35	3imtr4g 261	. 2 ⊢ ((F Fn A ∧ G Fn A) → (∀x ∈ A (F ‘x) = (G ‘x) → F = G))
37	2, 36	impbid2 195	1 ⊢ ((F Fn A ∧ G Fn A) → (F = G ↔ ∀x ∈ A (F ‘x) = (G ‘x)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ⟨cop 4562  dom cdm 4773   Fn wfn 4777   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  eqfnfv2  5394  eqfnfvd  5396  eqfnfv2f  5397  fvreseq  5399  fconst2g  5453