Theorem ffvresb 7080

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 5979	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
3		inss2 4192	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
4	2, 3	eqsstri 3982	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ dom 𝐹
5	1, 4	eqsstrrdi 3981	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
6	5	sselda 3935	. . . 4 ⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹)
7		fvres 6861	. . . . . 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 2838	. . . 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 3130	. 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 3075	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝐹)
15		dfss3 3924	. . . . . 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 582	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴) Fn 𝐴)
20	16, 19	sylan2 594	. . . 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 2822	. . . . . . . 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 3072	. . . . . 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 585	. . . 4 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)) → ran (𝐹 ↾ 𝐴) ⊆ 𝐵)
28		df-f 6504	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) ⊆ 𝐵))
29	20, 27, 28	sylanbrc 584	. . 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 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  dom cdm 5632  ran crn 5633   ↾ cres 5634  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by:  inlresf  9838  inrresf  9840  oppccatf  17663  lmbr2  23215  lmff  23257  lmmbr2  25227  iscau2  25245  relogbf  26769  sseqf  34570  rpsqrtcn  34771  climrescn  46106  climxrrelem  46107  climxrre  46108  liminflimsupxrre  46175  xlimxrre  46189  fourierdlem97  46561