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Theorem neificl 36621
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 770 . . 3 ((𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†) ∧ (𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…)) β†’ 𝑁 ∈ Fin)
2 innei 22629 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∩ 𝑦) ∈ ((neiβ€˜π½)β€˜π‘†))
323expib 1123 . . . . . . 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 3199 . . . . . 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 9324 . . . . . 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 4008 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ βŠ† ((neiβ€˜π½)β€˜π‘†) ↔ 𝑁 βŠ† ((neiβ€˜π½)β€˜π‘†)))
8 neeq1 3004 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ β‰  βˆ… ↔ 𝑁 β‰  βˆ…))
9 eleq1 2822 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ (π‘₯ ∈ Fin ↔ 𝑁 ∈ Fin))
107, 8, 93anbi123d 1437 . . . . . . . 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 1100 . . . . . . . . 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 1096 . . . . . . . . 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 275 . . . . . . . 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 287 . . . . . . 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 4954 . . . . . . . 8 (π‘₯ = 𝑁 β†’ ∩ π‘₯ = ∩ 𝑁)
1615eleq1d 2819 . . . . . . 7 (π‘₯ = 𝑁 β†’ (∩ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) ↔ ∩ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
1714, 16imbi12d 345 . . . . . 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 3587 . . . . 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 457 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 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  βˆ© cint 4951  β€˜cfv 6544  Fincfn 8939  Topctop 22395  neicnei 22601
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-top 22396  df-nei 22602
This theorem is referenced by: (None)
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