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Theorem gneispacef2 42877
Description: A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
Hypothesis
Ref Expression
gneispace.a 𝐴 = {𝑓 ∣ (𝑓:dom π‘“βŸΆ(𝒫 (𝒫 dom 𝑓 βˆ– {βˆ…}) βˆ– {βˆ…}) ∧ βˆ€π‘ ∈ dom π‘“βˆ€π‘› ∈ (π‘“β€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝑓(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (π‘“β€˜π‘))))}
Assertion
Ref Expression
gneispacef2 (𝐹 ∈ 𝐴 β†’ 𝐹:dom πΉβŸΆπ’« 𝒫 dom 𝐹)
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)

Proof of Theorem gneispacef2
StepHypRef Expression
1 elex 3492 . . . 4 (𝐹 ∈ 𝐴 β†’ 𝐹 ∈ V)
2 gneispace.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:dom π‘“βŸΆ(𝒫 (𝒫 dom 𝑓 βˆ– {βˆ…}) βˆ– {βˆ…}) ∧ βˆ€π‘ ∈ dom π‘“βˆ€π‘› ∈ (π‘“β€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝑓(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (π‘“β€˜π‘))))}
32gneispace 42875 . . . 4 (𝐹 ∈ V β†’ (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘)))))))
41, 3syl 17 . . 3 (𝐹 ∈ 𝐴 β†’ (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘)))))))
54ibi 266 . 2 (𝐹 ∈ 𝐴 β†’ (Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘))))))
6 simp1 1136 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘))))) β†’ Fun 𝐹)
76funfnd 6579 . . 3 ((Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘))))) β†’ 𝐹 Fn dom 𝐹)
8 simp2 1137 . . 3 ((Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘))))) β†’ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹)
9 df-f 6547 . . 3 (𝐹:dom πΉβŸΆπ’« 𝒫 dom 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹))
107, 8, 9sylanbrc 583 . 2 ((Fun 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹 ∧ βˆ€π‘ ∈ dom 𝐹((πΉβ€˜π‘) β‰  βˆ… ∧ βˆ€π‘› ∈ (πΉβ€˜π‘)(𝑝 ∈ 𝑛 ∧ βˆ€π‘  ∈ 𝒫 dom 𝐹(𝑛 βŠ† 𝑠 β†’ 𝑠 ∈ (πΉβ€˜π‘))))) β†’ 𝐹:dom πΉβŸΆπ’« 𝒫 dom 𝐹)
115, 10syl 17 1 (𝐹 ∈ 𝐴 β†’ 𝐹:dom πΉβŸΆπ’« 𝒫 dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by: (None)
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