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Theorem ffvresb 6558
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 6212 . . . . . 6 ((𝐹𝐴):𝐴𝐵 → dom (𝐹𝐴) = 𝐴)
2 dmres 5577 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
3 inss2 3977 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
42, 3eqsstri 3776 . . . . . 6 dom (𝐹𝐴) ⊆ dom 𝐹
51, 4syl6eqssr 3797 . . . . 5 ((𝐹𝐴):𝐴𝐵𝐴 ⊆ dom 𝐹)
65sselda 3744 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
7 fvres 6369 . . . . . 6 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
87adantl 473 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
9 ffvelrn 6521 . . . . 5 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → ((𝐹𝐴)‘𝑥) ∈ 𝐵)
108, 9eqeltrrd 2840 . . . 4 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
116, 10jca 555 . . 3 (((𝐹𝐴):𝐴𝐵𝑥𝐴) → (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
1211ralrimiva 3104 . 2 ((𝐹𝐴):𝐴𝐵 → ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))
13 simpl 474 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝑥 ∈ dom 𝐹)
1413ralimi 3090 . . . . . 6 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
15 dfss3 3733 . . . . . 6 (𝐴 ⊆ dom 𝐹 ↔ ∀𝑥𝐴 𝑥 ∈ dom 𝐹)
1614, 15sylibr 224 . . . . 5 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → 𝐴 ⊆ dom 𝐹)
17 funfn 6079 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnssres 6165 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) Fn 𝐴)
1917, 18sylanb 490 . . . . 5 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) Fn 𝐴)
2016, 19sylan2 492 . . . 4 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → (𝐹𝐴) Fn 𝐴)
21 simpr 479 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → (𝐹𝑥) ∈ 𝐵)
227eleq1d 2824 . . . . . . . 8 (𝑥𝐴 → (((𝐹𝐴)‘𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
2321, 22syl5ibr 236 . . . . . . 7 (𝑥𝐴 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐹𝐴)‘𝑥) ∈ 𝐵))
2423ralimia 3088 . . . . . 6 (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵)
2524adantl 473 . . . . 5 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵)
26 fnfvrnss 6554 . . . . 5 (((𝐹𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝐹𝐴)‘𝑥) ∈ 𝐵) → ran (𝐹𝐴) ⊆ 𝐵)
2720, 25, 26syl2anc 696 . . . 4 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → ran (𝐹𝐴) ⊆ 𝐵)
28 df-f 6053 . . . 4 ((𝐹𝐴):𝐴𝐵 ↔ ((𝐹𝐴) Fn 𝐴 ∧ ran (𝐹𝐴) ⊆ 𝐵))
2920, 27, 28sylanbrc 701 . . 3 ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)) → (𝐹𝐴):𝐴𝐵)
3029ex 449 . 2 (Fun 𝐹 → (∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵) → (𝐹𝐴):𝐴𝐵))
3112, 30impbid2 216 1 (Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715  dom cdm 5266  ran crn 5267  cres 5268  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  inlresf  8951  inrresf  8953  lmbr2  21285  lmff  21327  lmmbr2  23277  iscau2  23295  relogbf  24749  sseqf  30784  rpsqrtcn  31001  climrescn  40501  climxrrelem  40502  climxrre  40503  xlimxrre  40578  fourierdlem97  40941
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