Theorem eqfnfv2 5394

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 5-Feb-2004.)

Assertion

Ref	Expression
eqfnfv2	⊢ ((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

Allowed substitution hint:   B(x)

Proof of Theorem eqfnfv2

Step	Hyp	Ref	Expression
1		dmeq 4908	. . . 4 ⊢ (F = G → dom F = dom G)
2		fndm 5183	. . . . 5 ⊢ (F Fn A → dom F = A)
3		fndm 5183	. . . . 5 ⊢ (G Fn B → dom G = B)
4	2, 3	eqeqan12d 2368	. . . 4 ⊢ ((F Fn A ∧ G Fn B) → (dom F = dom G ↔ A = B))
5	1, 4	syl5ib 210	. . 3 ⊢ ((F Fn A ∧ G Fn B) → (F = G → A = B))
6	5	pm4.71rd 616	. 2 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ F = G)))
7		fneq2 5175	. . . . . 6 ⊢ (A = B → (G Fn A ↔ G Fn B))
8	7	biimparc 473	. . . . 5 ⊢ ((G Fn B ∧ A = B) → G Fn A)
9		eqfnfv 5393	. . . . 5 ⊢ ((F Fn A ∧ G Fn A) → (F = G ↔ ∀x ∈ A (F ‘x) = (G ‘x)))
10	8, 9	sylan2 460	. . . 4 ⊢ ((F Fn A ∧ (G Fn B ∧ A = B)) → (F = G ↔ ∀x ∈ A (F ‘x) = (G ‘x)))
11	10	anassrs 629	. . 3 ⊢ (((F Fn A ∧ G Fn B) ∧ A = B) → (F = G ↔ ∀x ∈ A (F ‘x) = (G ‘x)))
12	11	pm5.32da 622	. 2 ⊢ ((F Fn A ∧ G Fn B) → ((A = B ∧ F = G) ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))
13	6, 12	bitrd 244	1 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2615  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:  eqfnfv3  5395  eqfunfv  5398  eqfnov  5590