Theorem ffvresb 5432

Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
ffvresb	⊢ (Fun F → ((F ↾ A):A–→B ↔ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)))

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

Proof of Theorem ffvresb

Step	Hyp	Ref	Expression
1		fdm 5227	. . . . . 6 ⊢ ((F ↾ A):A–→B → dom (F ↾ A) = A)
2		dmres 4987	. . . . . . . 8 ⊢ dom (F ↾ A) = (A ∩ dom F)
3		inss2 3477	. . . . . . . 8 ⊢ (A ∩ dom F) ⊆ dom F
4	2, 3	eqsstri 3302	. . . . . . 7 ⊢ dom (F ↾ A) ⊆ dom F
5	4	a1i 10	. . . . . 6 ⊢ ((F ↾ A):A–→B → dom (F ↾ A) ⊆ dom F)
6	1, 5	eqsstr3d 3307	. . . . 5 ⊢ ((F ↾ A):A–→B → A ⊆ dom F)
7	6	sselda 3274	. . . 4 ⊢ (((F ↾ A):A–→B ∧ x ∈ A) → x ∈ dom F)
8		fvres 5343	. . . . . 6 ⊢ (x ∈ A → ((F ↾ A) ‘x) = (F ‘x))
9	8	adantl 452	. . . . 5 ⊢ (((F ↾ A):A–→B ∧ x ∈ A) → ((F ↾ A) ‘x) = (F ‘x))
10		ffvelrn 5416	. . . . 5 ⊢ (((F ↾ A):A–→B ∧ x ∈ A) → ((F ↾ A) ‘x) ∈ B)
11	9, 10	eqeltrrd 2428	. . . 4 ⊢ (((F ↾ A):A–→B ∧ x ∈ A) → (F ‘x) ∈ B)
12	7, 11	jca 518	. . 3 ⊢ (((F ↾ A):A–→B ∧ x ∈ A) → (x ∈ dom F ∧ (F ‘x) ∈ B))
13	12	ralrimiva 2698	. 2 ⊢ ((F ↾ A):A–→B → ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B))
14		simpl 443	. . . . . . 7 ⊢ ((x ∈ dom F ∧ (F ‘x) ∈ B) → x ∈ dom F)
15	14	ralimi 2690	. . . . . 6 ⊢ (∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B) → ∀x ∈ A x ∈ dom F)
16		dfss3 3264	. . . . . 6 ⊢ (A ⊆ dom F ↔ ∀x ∈ A x ∈ dom F)
17	15, 16	sylibr 203	. . . . 5 ⊢ (∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B) → A ⊆ dom F)
18		funfn 5137	. . . . . 6 ⊢ (Fun F ↔ F Fn dom F)
19		fnssres 5197	. . . . . 6 ⊢ ((F Fn dom F ∧ A ⊆ dom F) → (F ↾ A) Fn A)
20	18, 19	sylanb 458	. . . . 5 ⊢ ((Fun F ∧ A ⊆ dom F) → (F ↾ A) Fn A)
21	17, 20	sylan2 460	. . . 4 ⊢ ((Fun F ∧ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)) → (F ↾ A) Fn A)
22		simpr 447	. . . . . . . 8 ⊢ ((x ∈ dom F ∧ (F ‘x) ∈ B) → (F ‘x) ∈ B)
23	8	eleq1d 2419	. . . . . . . 8 ⊢ (x ∈ A → (((F ↾ A) ‘x) ∈ B ↔ (F ‘x) ∈ B))
24	22, 23	syl5ibr 212	. . . . . . 7 ⊢ (x ∈ A → ((x ∈ dom F ∧ (F ‘x) ∈ B) → ((F ↾ A) ‘x) ∈ B))
25	24	ralimia 2688	. . . . . 6 ⊢ (∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B) → ∀x ∈ A ((F ↾ A) ‘x) ∈ B)
26	25	adantl 452	. . . . 5 ⊢ ((Fun F ∧ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)) → ∀x ∈ A ((F ↾ A) ‘x) ∈ B)
27		fnfvrnss 5430	. . . . 5 ⊢ (((F ↾ A) Fn A ∧ ∀x ∈ A ((F ↾ A) ‘x) ∈ B) → ran (F ↾ A) ⊆ B)
28	21, 26, 27	syl2anc 642	. . . 4 ⊢ ((Fun F ∧ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)) → ran (F ↾ A) ⊆ B)
29		df-f 4792	. . . 4 ⊢ ((F ↾ A):A–→B ↔ ((F ↾ A) Fn A ∧ ran (F ↾ A) ⊆ B))
30	21, 28, 29	sylanbrc 645	. . 3 ⊢ ((Fun F ∧ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)) → (F ↾ A):A–→B)
31	30	ex 423	. 2 ⊢ (Fun F → (∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B) → (F ↾ A):A–→B))
32	13, 31	impbid2 195	1 ⊢ (Fun F → ((F ↾ A):A–→B ↔ ∀x ∈ A (x ∈ dom F ∧ (F ‘x) ∈ B)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ∩ cin 3209   ⊆ wss 3258  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776   Fn wfn 4777  –→wf 4778   ‘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-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796

This theorem is referenced by: (None)