Theorem ffvresb 5271

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 4993	. . . . . 6 ⊢ ((𝐹 ↾ A):A⟶B → dom (𝐹 ↾ A) = A)
2		dmres 4575	. . . . . . 7 ⊢ dom (𝐹 ↾ A) = (A ∩ dom 𝐹)
3		inss2 3152	. . . . . . 7 ⊢ (A ∩ dom 𝐹) ⊆ dom 𝐹
4	2, 3	eqsstri 2969	. . . . . 6 ⊢ dom (𝐹 ↾ A) ⊆ dom 𝐹
5	1, 4	syl6eqssr 2990	. . . . 5 ⊢ ((𝐹 ↾ A):A⟶B → A ⊆ dom 𝐹)
6	5	sselda 2939	. . . 4 ⊢ (((𝐹 ↾ A):A⟶B ∧ x ∈ A) → x ∈ dom 𝐹)
7		fvres 5141	. . . . . 6 ⊢ (x ∈ A → ((𝐹 ↾ A)‘x) = (𝐹‘x))
8	7	adantl 262	. . . . 5 ⊢ (((𝐹 ↾ A):A⟶B ∧ x ∈ A) → ((𝐹 ↾ A)‘x) = (𝐹‘x))
9		ffvelrn 5243	. . . . 5 ⊢ (((𝐹 ↾ A):A⟶B ∧ x ∈ A) → ((𝐹 ↾ A)‘x) ∈ B)
10	8, 9	eqeltrrd 2112	. . . 4 ⊢ (((𝐹 ↾ A):A⟶B ∧ x ∈ A) → (𝐹‘x) ∈ B)
11	6, 10	jca 290	. . 3 ⊢ (((𝐹 ↾ A):A⟶B ∧ x ∈ A) → (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B))
12	11	ralrimiva 2386	. 2 ⊢ ((𝐹 ↾ A):A⟶B → ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B))
13		simpl 102	. . . . . . 7 ⊢ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → x ∈ dom 𝐹)
14	13	ralimi 2378	. . . . . 6 ⊢ (∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → ∀x ∈ A x ∈ dom 𝐹)
15		dfss3 2929	. . . . . 6 ⊢ (A ⊆ dom 𝐹 ↔ ∀x ∈ A x ∈ dom 𝐹)
16	14, 15	sylibr 137	. . . . 5 ⊢ (∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → A ⊆ dom 𝐹)
17		funfn 4874	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnssres 4955	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ A ⊆ dom 𝐹) → (𝐹 ↾ A) Fn A)
19	17, 18	sylanb 268	. . . . 5 ⊢ ((Fun 𝐹 ∧ A ⊆ dom 𝐹) → (𝐹 ↾ A) Fn A)
20	16, 19	sylan2 270	. . . 4 ⊢ ((Fun 𝐹 ∧ ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B)) → (𝐹 ↾ A) Fn A)
21		simpr 103	. . . . . . . 8 ⊢ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → (𝐹‘x) ∈ B)
22	7	eleq1d 2103	. . . . . . . 8 ⊢ (x ∈ A → (((𝐹 ↾ A)‘x) ∈ B ↔ (𝐹‘x) ∈ B))
23	21, 22	syl5ibr 145	. . . . . . 7 ⊢ (x ∈ A → ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → ((𝐹 ↾ A)‘x) ∈ B))
24	23	ralimia 2376	. . . . . 6 ⊢ (∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → ∀x ∈ A ((𝐹 ↾ A)‘x) ∈ B)
25	24	adantl 262	. . . . 5 ⊢ ((Fun 𝐹 ∧ ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B)) → ∀x ∈ A ((𝐹 ↾ A)‘x) ∈ B)
26		fnfvrnss 5268	. . . . 5 ⊢ (((𝐹 ↾ A) Fn A ∧ ∀x ∈ A ((𝐹 ↾ A)‘x) ∈ B) → ran (𝐹 ↾ A) ⊆ B)
27	20, 25, 26	syl2anc 391	. . . 4 ⊢ ((Fun 𝐹 ∧ ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B)) → ran (𝐹 ↾ A) ⊆ B)
28		df-f 4849	. . . 4 ⊢ ((𝐹 ↾ A):A⟶B ↔ ((𝐹 ↾ A) Fn A ∧ ran (𝐹 ↾ A) ⊆ B))
29	20, 27, 28	sylanbrc 394	. . 3 ⊢ ((Fun 𝐹 ∧ ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B)) → (𝐹 ↾ A):A⟶B)
30	29	ex 108	. 2 ⊢ (Fun 𝐹 → (∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B) → (𝐹 ↾ A):A⟶B))
31	12, 30	impbid2 131	1 ⊢ (Fun 𝐹 → ((𝐹 ↾ A):A⟶B ↔ ∀x ∈ A (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ B)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910   ⊆ wss 2911  dom cdm 4288  ran crn 4289   ↾ cres 4290  Fun wfun 4839   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853

This theorem is referenced by: (None)