Theorem fvreseq 5398

Description: Equality of restricted functions is determined by their values. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 6-Feb-2004.)

Assertion

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

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

Allowed substitution hint:   A(x)

Proof of Theorem fvreseq

Step	Hyp	Ref	Expression
1		fnssres 5196	. . . 4 ⊢ ((F Fn A ∧ B ⊆ A) → (F ↾ B) Fn B)
2		fnssres 5196	. . . 4 ⊢ ((G Fn A ∧ B ⊆ A) → (G ↾ B) Fn B)
3	1, 2	anim12i 549	. . 3 ⊢ (((F Fn A ∧ B ⊆ A) ∧ (G Fn A ∧ B ⊆ A)) → ((F ↾ B) Fn B ∧ (G ↾ B) Fn B))
4	3	anandirs 804	. 2 ⊢ (((F Fn A ∧ G Fn A) ∧ B ⊆ A) → ((F ↾ B) Fn B ∧ (G ↾ B) Fn B))
5		eqfnfv 5392	. . 3 ⊢ (((F ↾ B) Fn B ∧ (G ↾ B) Fn B) → ((F ↾ B) = (G ↾ B) ↔ ∀x ∈ B ((F ↾ B) ‘x) = ((G ↾ B) ‘x)))
6		fvres 5342	. . . . 5 ⊢ (x ∈ B → ((F ↾ B) ‘x) = (F ‘x))
7		fvres 5342	. . . . 5 ⊢ (x ∈ B → ((G ↾ B) ‘x) = (G ‘x))
8	6, 7	eqeq12d 2367	. . . 4 ⊢ (x ∈ B → (((F ↾ B) ‘x) = ((G ↾ B) ‘x) ↔ (F ‘x) = (G ‘x)))
9	8	ralbiia 2646	. . 3 ⊢ (∀x ∈ B ((F ↾ B) ‘x) = ((G ↾ B) ‘x) ↔ ∀x ∈ B (F ‘x) = (G ‘x))
10	5, 9	syl6bb 252	. 2 ⊢ (((F ↾ B) Fn B ∧ (G ↾ B) Fn B) → ((F ↾ B) = (G ↾ B) ↔ ∀x ∈ B (F ‘x) = (G ‘x)))
11	4, 10	syl 15	1 ⊢ (((F Fn A ∧ G Fn A) ∧ B ⊆ A) → ((F ↾ B) = (G ↾ B) ↔ ∀x ∈ B (F ‘x) = (G ‘x)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257   ↾ cres 4774   Fn wfn 4776   ‘cfv 4781

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by: (None)