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Theorem neificl 36616
Description: Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
Assertion
Ref Expression
neificl (((𝐽 ∈ Top ∧ 𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))

Proof of Theorem neificl
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 769 . . 3 ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ 𝑁 ∈ Fin)
2 innei 22628 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
323expib 1122 . . . . . . 7 (𝐽 ∈ Top β†’ ((π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†)))
43ralrimivv 3198 . . . . . 6 (𝐽 ∈ Top β†’ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)(π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
5 fiint 9323 . . . . . 6 (βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜π‘†)(π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘₯((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
64, 5sylib 217 . . . . 5 (𝐽 ∈ Top β†’ βˆ€π‘₯((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)))
7 sseq1 4007 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ↔ 𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†)))
8 neeq1 3003 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ β‰  βˆ… ↔ 𝑁 β‰  βˆ…))
9 eleq1 2821 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ ∈ Fin ↔ 𝑁 ∈ Fin))
107, 8, 93anbi123d 1436 . . . . . . . 8 (π‘₯ = 𝑁 β†’ ((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) ↔ (𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ 𝑁 β‰  βˆ… ∧ 𝑁 ∈ Fin)))
11 3ancomb 1099 . . . . . . . . 9 ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ 𝑁 β‰  βˆ… ∧ 𝑁 ∈ Fin) ↔ (𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…))
12 3anass 1095 . . . . . . . . 9 ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…) ↔ (𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)))
1311, 12bitri 274 . . . . . . . 8 ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ 𝑁 β‰  βˆ… ∧ 𝑁 ∈ Fin) ↔ (𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)))
1410, 13bitrdi 286 . . . . . . 7 (π‘₯ = 𝑁 β†’ ((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) ↔ (𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…))))
15 inteq 4953 . . . . . . . 8 (π‘₯ = 𝑁 β†’ ∩ π‘₯ = ∩ 𝑁)
1615eleq1d 2818 . . . . . . 7 (π‘₯ = 𝑁 β†’ (∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ↔ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
1714, 16imbi12d 344 . . . . . 6 (π‘₯ = 𝑁 β†’ (((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) ↔ ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))))
1817spcgv 3586 . . . . 5 (𝑁 ∈ Fin β†’ (βˆ€π‘₯((π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))))
196, 18syl5 34 . . . 4 (𝑁 ∈ Fin β†’ (𝐽 ∈ Top β†’ ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))))
2019com3l 89 . . 3 (𝐽 ∈ Top β†’ ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ (𝑁 ∈ Fin β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))))
211, 20mpdi 45 . 2 (𝐽 ∈ Top β†’ ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
2221impl 456 1 (((𝐽 ∈ Top ∧ 𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βˆ© cint 4950  β€˜cfv 6543  Fincfn 8938  Topctop 22394  neicnei 22600
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-er 8702  df-en 8939  df-fin 8942  df-top 22395  df-nei 22601
This theorem is referenced by: (None)
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