Theorem ffvresb 7079

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 6679	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → dom (𝐹 ↾ 𝐴) = 𝐴)
2		dmres 5972	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
3		inss2 4197	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
4	2, 3	eqsstri 3990	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
5	1, 4	eqsstrrdi 3989	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
6	5	sselda 3943	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
7		fvres 6859	. . . . . 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 7035	. . . . 5 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ 𝐵)
10	8, 9	eqeltrrd 2829	. . . 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 3125	. 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 3066	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹)
15		dfss3 3932	. . . . . 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 6530	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnssres 6623	. . . . . 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 2813	. . . . . . . 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 3063	. . . . . 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 7075	. . . . 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 6503	. . . 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 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3910   ⊆ wss 3911  dom cdm 5631  ran crn 5632   ↾ cres 5633  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  inlresf  9843  inrresf  9845  oppccatf  17665  lmbr2  23122  lmff  23164  lmmbr2  25135  iscau2  25153  relogbf  26677  sseqf  34356  rpsqrtcn  34557  climrescn  45719  climxrrelem  45720  climxrre  45721  liminflimsupxrre  45788  xlimxrre  45802  fourierdlem97  46174