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Theorem gneispacef2 42500
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 3465 . . . 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 42498 . . . 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 267 . 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 1137 . . . 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 6536 . . 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 1138 . . 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 6504 . . 3 (𝐹:dom πΉβŸΆπ’« 𝒫 dom 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 βŠ† 𝒫 𝒫 dom 𝐹))
107, 8, 9sylanbrc 584 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆ€wral 3061  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  {csn 4590  dom cdm 5637  ran crn 5638  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by: (None)
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