Theorem ffvresb 7068

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 6669	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → dom (𝐹 ↾ 𝐴) = 𝐴)
2		dmres 5969	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
3		inss2 4188	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
4	2, 3	eqsstri 3978	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
5	1, 4	eqsstrrdi 3977	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
6	5	sselda 3931	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
7		fvres 6851	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
8	7	adantl 481	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
9		ffvelcdm 7024	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)
10	8, 9	eqeltrrd 2835	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
11	6, 10	jca 511	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))
12	11	ralrimiva 3126	. 2 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))
13		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹)
14	13	ralimi 3071	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹)
15		dfss3 3920	. . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹)
16	14, 15	sylibr 234	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹)
17		funfn 6520	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnssres 6613	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴)
19	17, 18	sylanb 581	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴)
20	16, 19	sylan2 593	. . . 4 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴)
21		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹‘𝑥) ∈ 𝐵)
22	7	eleq1d 2819	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
23	21, 22	imbitrrid 246	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵))
24	23	ralimia 3068	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)
25	24	adantl 481	. . . . 5 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)
26		fnfvrnss 7064	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵)
27	20, 25, 26	syl2anc 584	. . . 4 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵)
28		df-f 6494	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) ⊆ 𝐵))
29	20, 27, 28	sylanbrc 583	. . 3 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
30	29	ex 412	. 2 ⊢ (Fun 𝐹 → (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐹 ↾ 𝐴):𝐴⟶𝐵))
31	12, 30	impbid2 226	1 ⊢ (Fun 𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∩ cin 3898   ⊆ wss 3899  dom cdm 5622  ran crn 5623   ↾ cres 5624  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498

This theorem is referenced by:  inlresf  9824  inrresf  9826  oppccatf  17649  lmbr2  23201  lmff  23243  lmmbr2  25213  iscau2  25231  relogbf  26755  sseqf  34498  rpsqrtcn  34699  climrescn  45934  climxrrelem  45935  climxrre  45936  liminflimsupxrre  46003  xlimxrre  46017  fourierdlem97  46389