Theorem eqfnfv3 5395

Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
eqfnfv3	⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x)))))

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

Proof of Theorem eqfnfv3

Step	Hyp	Ref	Expression
1		eqfnfv2 5394	. 2 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))
2		eqss 3288	. . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
3		ancom 437	. . . . 5 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (B ⊆ A ∧ A ⊆ B))
4	2, 3	bitri 240	. . . 4 ⊢ (A = B ↔ (B ⊆ A ∧ A ⊆ B))
5	4	anbi1i 676	. . 3 ⊢ ((A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x)) ↔ ((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (F ‘x) = (G ‘x)))
6		anass 630	. . 3 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (F ‘x) = (G ‘x)) ↔ (B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))
7		dfss3 3264	. . . . . 6 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)
8	7	anbi1i 676	. . . . 5 ⊢ ((A ⊆ B ∧ ∀x ∈ A (F ‘x) = (G ‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (F ‘x) = (G ‘x)))
9		r19.26 2747	. . . . 5 ⊢ (∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (F ‘x) = (G ‘x)))
10	8, 9	bitr4i 243	. . . 4 ⊢ ((A ⊆ B ∧ ∀x ∈ A (F ‘x) = (G ‘x)) ↔ ∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x)))
11	10	anbi2i 675	. . 3 ⊢ ((B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (F ‘x) = (G ‘x))) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x))))
12	5, 6, 11	3bitri 262	. 2 ⊢ ((A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x)) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x))))
13	1, 12	syl6bb 252	1 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (F ‘x) = (G ‘x)))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258   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: (None)