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Theorem opnfbas 22450
Description: The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
Hypothesis
Ref Expression
opnfbas.1 𝑋 = 𝐽
Assertion
Ref Expression
opnfbas ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem opnfbas
Dummy variables 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4042 . . . 4 {𝑥𝐽𝑆𝑥} ⊆ 𝐽
2 opnfbas.1 . . . . . 6 𝑋 = 𝐽
32eqimss2i 4012 . . . . 5 𝐽𝑋
4 sspwuni 5008 . . . . 5 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
53, 4mpbir 234 . . . 4 𝐽 ⊆ 𝒫 𝑋
61, 5sstri 3962 . . 3 {𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋
76a1i 11 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋)
82topopn 21514 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
98anim1i 617 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋𝐽𝑆𝑋))
1093adant3 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑋𝐽𝑆𝑋))
11 sseq2 3979 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1211elrab 3666 . . . . 5 (𝑋 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑋𝐽𝑆𝑋))
1310, 12sylibr 237 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ {𝑥𝐽𝑆𝑥})
1413ne0d 4284 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ≠ ∅)
15 ss0 4335 . . . . . . 7 (𝑆 ⊆ ∅ → 𝑆 = ∅)
1615necon3ai 3039 . . . . . 6 (𝑆 ≠ ∅ → ¬ 𝑆 ⊆ ∅)
17163ad2ant3 1132 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ 𝑆 ⊆ ∅)
1817intnand 492 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ (∅ ∈ 𝐽𝑆 ⊆ ∅))
19 df-nel 3119 . . . . 5 (∅ ∉ {𝑥𝐽𝑆𝑥} ↔ ¬ ∅ ∈ {𝑥𝐽𝑆𝑥})
20 sseq2 3979 . . . . . . 7 (𝑥 = ∅ → (𝑆𝑥𝑆 ⊆ ∅))
2120elrab 3666 . . . . . 6 (∅ ∈ {𝑥𝐽𝑆𝑥} ↔ (∅ ∈ 𝐽𝑆 ⊆ ∅))
2221notbii 323 . . . . 5 (¬ ∅ ∈ {𝑥𝐽𝑆𝑥} ↔ ¬ (∅ ∈ 𝐽𝑆 ⊆ ∅))
2319, 22bitr2i 279 . . . 4 (¬ (∅ ∈ 𝐽𝑆 ⊆ ∅) ↔ ∅ ∉ {𝑥𝐽𝑆𝑥})
2418, 23sylib 221 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ∅ ∉ {𝑥𝐽𝑆𝑥})
25 sseq2 3979 . . . . . . 7 (𝑥 = 𝑟 → (𝑆𝑥𝑆𝑟))
2625elrab 3666 . . . . . 6 (𝑟 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑟𝐽𝑆𝑟))
27 sseq2 3979 . . . . . . 7 (𝑥 = 𝑠 → (𝑆𝑥𝑆𝑠))
2827elrab 3666 . . . . . 6 (𝑠 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑠𝐽𝑆𝑠))
2926, 28anbi12i 629 . . . . 5 ((𝑟 ∈ {𝑥𝐽𝑆𝑥} ∧ 𝑠 ∈ {𝑥𝐽𝑆𝑥}) ↔ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)))
30 simpl 486 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝐽 ∈ Top)
31 simprll 778 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑟𝐽)
32 simprrl 780 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑠𝐽)
33 inopn 21507 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑟𝐽𝑠𝐽) → (𝑟𝑠) ∈ 𝐽)
3430, 31, 32, 33syl3anc 1368 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → (𝑟𝑠) ∈ 𝐽)
35 ssin 4192 . . . . . . . . . . . . 13 ((𝑆𝑟𝑆𝑠) ↔ 𝑆 ⊆ (𝑟𝑠))
3635biimpi 219 . . . . . . . . . . . 12 ((𝑆𝑟𝑆𝑠) → 𝑆 ⊆ (𝑟𝑠))
3736ad2ant2l 745 . . . . . . . . . . 11 (((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)) → 𝑆 ⊆ (𝑟𝑠))
3837adantl 485 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑆 ⊆ (𝑟𝑠))
3934, 38jca 515 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
40393ad2antl1 1182 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
41 sseq2 3979 . . . . . . . . 9 (𝑥 = (𝑟𝑠) → (𝑆𝑥𝑆 ⊆ (𝑟𝑠)))
4241elrab 3666 . . . . . . . 8 ((𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥} ↔ ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
4340, 42sylibr 237 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → (𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥})
44 ssid 3975 . . . . . . 7 (𝑟𝑠) ⊆ (𝑟𝑠)
45 sseq1 3978 . . . . . . . 8 (𝑡 = (𝑟𝑠) → (𝑡 ⊆ (𝑟𝑠) ↔ (𝑟𝑠) ⊆ (𝑟𝑠)))
4645rspcev 3609 . . . . . . 7 (((𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥} ∧ (𝑟𝑠) ⊆ (𝑟𝑠)) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
4743, 44, 46sylancl 589 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
4847ex 416 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
4929, 48syl5bi 245 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑟 ∈ {𝑥𝐽𝑆𝑥} ∧ 𝑠 ∈ {𝑥𝐽𝑆𝑥}) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
5049ralrimivv 3185 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
5114, 24, 503jca 1125 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
52 isfbas2 22443 . . . 4 (𝑋𝐽 → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
538, 52syl 17 . . 3 (𝐽 ∈ Top → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
54533ad2ant1 1130 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
557, 51, 54mpbir2and 712 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wnel 3118  wral 3133  wrex 3134  {crab 3137  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522   cuni 4824  cfv 6343  fBascfbas 20533  Topctop 21501
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-fbas 20542  df-top 21502
This theorem is referenced by:  neifg  33776
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