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Theorem ffvresb 7145
Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
ffvresb (Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem ffvresb
StepHypRef Expression
1 fdm 6746 . . . . . 6 ((𝐹𝐴):𝐴𝐵 → dom (𝐹𝐴) = 𝐴)
2 dmres 6032 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
3 inss2 4246 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
42, 3eqsstri 4030 . . . . . 6 dom (𝐹𝐴) ⊆ dom 𝐹
51, 4eqsstrrdi 4051 . . . . 5 ((𝐹𝐴):𝐴𝐵𝐴 ⊆ dom 𝐹)
65sselda 3995 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
7 fvres 6926 . . . . . 6 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
87adantl 481 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
9 ffvelcdm 7101 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) ∈ 𝐵)
108, 9eqeltrrd 2840 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
116, 10jca 511 . . 3 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
1211ralrimiva 3144 . 2 ((𝐹𝐴):𝐴𝐵 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
13 simpl 482 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹)
1413ralimi 3081 . . . . . 6 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
15 dfss3 3984 . . . . . 6 (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
1614, 15sylibr 234 . . . . 5 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹)
17 funfn 6598 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnssres 6692 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) Fn 𝐴)
1917, 18sylanb 581 . . . . 5 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) Fn 𝐴)
2016, 19sylan2 593 . . . 4 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → (𝐹𝐴) Fn 𝐴)
21 simpr 484 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ 𝐵)
227eleq1d 2824 . . . . . . . 8 (𝑥𝐴 → (((𝐹𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
2321, 22imbitrrid 246 . . . . . . 7 (𝑥𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐹𝐴)‘𝑥) ∈ 𝐵))
2423ralimia 3078 . . . . . 6 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵)
2524adantl 481 . . . . 5 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵)
26 fnfvrnss 7141 . . . . 5 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵) → ran (𝐹𝐴) ⊆ 𝐵)
2720, 25, 26syl2anc 584 . . . 4 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → ran (𝐹𝐴) ⊆ 𝐵)
28 df-f 6567 . . . 4 ((𝐹𝐴):𝐴𝐵 ↔ ((𝐹𝐴) Fn 𝐴 ∧ ran (𝐹𝐴) ⊆ 𝐵))
2920, 27, 28sylanbrc 583 . . 3 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → (𝐹𝐴):𝐴𝐵)
3029ex 412 . 2 (Fun 𝐹 → (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴):𝐴𝐵))
3112, 30impbid2 226 1 (Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  dom cdm 5689  ran crn 5690  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  inlresf  9952  inrresf  9954  oppccatf  17775  lmbr2  23283  lmff  23325  lmmbr2  25307  iscau2  25325  relogbf  26849  sseqf  34374  rpsqrtcn  34587  climrescn  45704  climxrrelem  45705  climxrre  45706  liminflimsupxrre  45773  xlimxrre  45787  fourierdlem97  46159
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