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Theorem opnnei 21124
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei
StepHypRef Expression
1 0opn 20909 . . . . 5 (𝐽 ∈ Top → ∅ ∈ 𝐽)
21adantr 472 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3 eleq1 2825 . . . . 5 (𝑆 = ∅ → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
43adantl 473 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
52, 4mpbird 247 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆𝐽)
6 rzal 4215 . . . 4 (𝑆 = ∅ → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
76adantl 473 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
85, 72thd 255 . 2 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9 opnneip 21123 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝐽𝑥𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1093expia 1115 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑥𝑆𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1110ralrimiv 3101 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝐽) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1211ex 449 . . . 4 (𝐽 ∈ Top → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1312adantr 472 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14 df-ne 2931 . . . . . 6 (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15 r19.2z 4202 . . . . . . 7 ((𝑆 ≠ ∅ ∧ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1615ex 449 . . . . . 6 (𝑆 ≠ ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1714, 16sylbir 225 . . . . 5 𝑆 = ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18 eqid 2758 . . . . . . . 8 𝐽 = 𝐽
1918neii1 21110 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 𝐽)
2019ex 449 . . . . . 6 (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2120rexlimdvw 3170 . . . . 5 (𝐽 ∈ Top → (∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2217, 21sylan9r 693 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2318ntrss2 21061 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
2423adantr 472 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25 vex 3341 . . . . . . . . . . . . 13 𝑥 ∈ V
2625snss 4458 . . . . . . . . . . . 12 (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
2726ralbii 3116 . . . . . . . . . . 11 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28 dfss3 3731 . . . . . . . . . . . . 13 (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
2928biimpri 218 . . . . . . . . . . . 12 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3029adantl 473 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3127, 30sylan2br 494 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3224, 31eqssd 3759 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
3332ex 449 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
3425snss 4458 . . . . . . . . . . . 12 (𝑥𝑆 ↔ {𝑥} ⊆ 𝑆)
35 sstr2 3749 . . . . . . . . . . . . . 14 ({𝑥} ⊆ 𝑆 → (𝑆 𝐽 → {𝑥} ⊆ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3736adantl 473 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3834, 37syl5bi 232 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑥𝑆 → {𝑥} ⊆ 𝐽))
3938imp 444 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → {𝑥} ⊆ 𝐽)
4018neiint 21108 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝐽𝑆 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41403com23 1121 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42413expa 1112 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4339, 42syldan 488 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4443ralbidva 3121 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4518isopn3 21070 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
4633, 44, 453imtr4d 283 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
4746ex 449 . . . . . 6 (𝐽 ∈ Top → (𝑆 𝐽 → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
4948adantr 472 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
5113, 50impbid 202 . 2 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
528, 51pm2.61dan 867 1 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wne 2930  wral 3048  wrex 3049  wss 3713  c0 4056  {csn 4319   cuni 4586  cfv 6047  Topctop 20898  intcnt 21021  neicnei 21101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-top 20899  df-ntr 21024  df-nei 21102
This theorem is referenced by:  neiptopreu  21137  flimcf  21985
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