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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 6745 . . . . . 6 ((𝐹𝐴):𝐴𝐵 → dom (𝐹𝐴) = 𝐴)
2 dmres 6030 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
3 inss2 4238 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
42, 3eqsstri 4030 . . . . . 6 dom (𝐹𝐴) ⊆ dom 𝐹
51, 4eqsstrrdi 4029 . . . . 5 ((𝐹𝐴):𝐴𝐵𝐴 ⊆ dom 𝐹)
65sselda 3983 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
7 fvres 6925 . . . . . 6 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
87adantl 481 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
9 ffvelcdm 7101 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) ∈ 𝐵)
108, 9eqeltrrd 2842 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
116, 10jca 511 . . 3 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
1211ralrimiva 3146 . 2 ((𝐹𝐴):𝐴𝐵 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
13 simpl 482 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹)
1413ralimi 3083 . . . . . 6 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
15 dfss3 3972 . . . . . 6 (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
1614, 15sylibr 234 . . . . 5 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹)
17 funfn 6596 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnssres 6691 . . . . . 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 2826 . . . . . . . 8 (𝑥𝐴 → (((𝐹𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
2321, 22imbitrrid 246 . . . . . . 7 (𝑥𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐹𝐴)‘𝑥) ∈ 𝐵))
2423ralimia 3080 . . . . . 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 6565 . . . 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 1540  wcel 2108  wral 3061  cin 3950  wss 3951  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  inlresf  9954  inrresf  9956  oppccatf  17771  lmbr2  23267  lmff  23309  lmmbr2  25293  iscau2  25311  relogbf  26834  sseqf  34394  rpsqrtcn  34608  climrescn  45763  climxrrelem  45764  climxrre  45765  liminflimsupxrre  45832  xlimxrre  45846  fourierdlem97  46218
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