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Theorem opnfbas 21569
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 3671 . . . 4 {𝑥𝐽𝑆𝑥} ⊆ 𝐽
2 opnfbas.1 . . . . . 6 𝑋 = 𝐽
32eqimss2i 3644 . . . . 5 𝐽𝑋
4 sspwuni 4582 . . . . 5 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
53, 4mpbir 221 . . . 4 𝐽 ⊆ 𝒫 𝑋
61, 5sstri 3596 . . 3 {𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋
76a1i 11 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋)
82topopn 20643 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
98anim1i 591 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋𝐽𝑆𝑋))
1093adant3 1079 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑋𝐽𝑆𝑋))
11 sseq2 3611 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1211elrab 3350 . . . . 5 (𝑋 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑋𝐽𝑆𝑋))
1310, 12sylibr 224 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝑋 ∈ {𝑥𝐽𝑆𝑥})
14 ne0i 3902 . . . 4 (𝑋 ∈ {𝑥𝐽𝑆𝑥} → {𝑥𝐽𝑆𝑥} ≠ ∅)
1513, 14syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ≠ ∅)
16 ss0 3951 . . . . . . 7 (𝑆 ⊆ ∅ → 𝑆 = ∅)
1716necon3ai 2815 . . . . . 6 (𝑆 ≠ ∅ → ¬ 𝑆 ⊆ ∅)
18173ad2ant3 1082 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ 𝑆 ⊆ ∅)
1918intnand 961 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ¬ (∅ ∈ 𝐽𝑆 ⊆ ∅))
20 df-nel 2894 . . . . 5 (∅ ∉ {𝑥𝐽𝑆𝑥} ↔ ¬ ∅ ∈ {𝑥𝐽𝑆𝑥})
21 sseq2 3611 . . . . . . 7 (𝑥 = ∅ → (𝑆𝑥𝑆 ⊆ ∅))
2221elrab 3350 . . . . . 6 (∅ ∈ {𝑥𝐽𝑆𝑥} ↔ (∅ ∈ 𝐽𝑆 ⊆ ∅))
2322notbii 310 . . . . 5 (¬ ∅ ∈ {𝑥𝐽𝑆𝑥} ↔ ¬ (∅ ∈ 𝐽𝑆 ⊆ ∅))
2420, 23bitr2i 265 . . . 4 (¬ (∅ ∈ 𝐽𝑆 ⊆ ∅) ↔ ∅ ∉ {𝑥𝐽𝑆𝑥})
2519, 24sylib 208 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ∅ ∉ {𝑥𝐽𝑆𝑥})
26 sseq2 3611 . . . . . . 7 (𝑥 = 𝑟 → (𝑆𝑥𝑆𝑟))
2726elrab 3350 . . . . . 6 (𝑟 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑟𝐽𝑆𝑟))
28 sseq2 3611 . . . . . . 7 (𝑥 = 𝑠 → (𝑆𝑥𝑆𝑠))
2928elrab 3350 . . . . . 6 (𝑠 ∈ {𝑥𝐽𝑆𝑥} ↔ (𝑠𝐽𝑆𝑠))
3027, 29anbi12i 732 . . . . 5 ((𝑟 ∈ {𝑥𝐽𝑆𝑥} ∧ 𝑠 ∈ {𝑥𝐽𝑆𝑥}) ↔ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)))
31 simpl 473 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝐽 ∈ Top)
32 simprll 801 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑟𝐽)
33 simprrl 803 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑠𝐽)
34 inopn 20636 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑟𝐽𝑠𝐽) → (𝑟𝑠) ∈ 𝐽)
3531, 32, 33, 34syl3anc 1323 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → (𝑟𝑠) ∈ 𝐽)
36 ssin 3818 . . . . . . . . . . . . 13 ((𝑆𝑟𝑆𝑠) ↔ 𝑆 ⊆ (𝑟𝑠))
3736biimpi 206 . . . . . . . . . . . 12 ((𝑆𝑟𝑆𝑠) → 𝑆 ⊆ (𝑟𝑠))
3837ad2ant2l 781 . . . . . . . . . . 11 (((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)) → 𝑆 ⊆ (𝑟𝑠))
3938adantl 482 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → 𝑆 ⊆ (𝑟𝑠))
4035, 39jca 554 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
41403ad2antl1 1221 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
42 sseq2 3611 . . . . . . . . 9 (𝑥 = (𝑟𝑠) → (𝑆𝑥𝑆 ⊆ (𝑟𝑠)))
4342elrab 3350 . . . . . . . 8 ((𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥} ↔ ((𝑟𝑠) ∈ 𝐽𝑆 ⊆ (𝑟𝑠)))
4441, 43sylibr 224 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → (𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥})
45 ssid 3608 . . . . . . 7 (𝑟𝑠) ⊆ (𝑟𝑠)
46 sseq1 3610 . . . . . . . 8 (𝑡 = (𝑟𝑠) → (𝑡 ⊆ (𝑟𝑠) ↔ (𝑟𝑠) ⊆ (𝑟𝑠)))
4746rspcev 3298 . . . . . . 7 (((𝑟𝑠) ∈ {𝑥𝐽𝑆𝑥} ∧ (𝑟𝑠) ⊆ (𝑟𝑠)) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
4844, 45, 47sylancl 693 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) ∧ ((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠))) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
4948ex 450 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (((𝑟𝐽𝑆𝑟) ∧ (𝑠𝐽𝑆𝑠)) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
5030, 49syl5bi 232 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ((𝑟 ∈ {𝑥𝐽𝑆𝑥} ∧ 𝑠 ∈ {𝑥𝐽𝑆𝑥}) → ∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
5150ralrimivv 2965 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠))
5215, 25, 513jca 1240 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))
53 isfbas2 21562 . . . 4 (𝑋𝐽 → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
548, 53syl 17 . . 3 (𝐽 ∈ Top → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
55543ad2ant1 1080 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → ({𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥𝐽𝑆𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥𝐽𝑆𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥𝐽𝑆𝑥} ∧ ∀𝑟 ∈ {𝑥𝐽𝑆𝑥}∀𝑠 ∈ {𝑥𝐽𝑆𝑥}∃𝑡 ∈ {𝑥𝐽𝑆𝑥}𝑡 ⊆ (𝑟𝑠)))))
567, 52, 55mpbir2and 956 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wnel 2893  wral 2907  wrex 2908  {crab 2911  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135   cuni 4407  cfv 5852  fBascfbas 19666  Topctop 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fv 5860  df-fbas 19675  df-top 20631
This theorem is referenced by:  neifg  32043
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