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Theorem neiint 12351
 Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1 𝑋 = 𝐽
Assertion
Ref Expression
neiint ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Proof of Theorem neiint
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = 𝐽
21isnei 12350 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
323adant3 1002 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
433anibar 1150 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
5 simprrl 529 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆𝑣)
61ssntr 12328 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
763adantl2 1139 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
87adantrrl 478 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
95, 8sstrd 3111 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
109rexlimdvaa 2553 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11 simpl1 985 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12 simpl3 987 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁𝑋)
131ntropn 12323 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
1411, 12, 13syl2anc 409 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15 simpr 109 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
161ntrss2 12327 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
1711, 12, 16syl2anc 409 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18 sseq2 3125 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆𝑣𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19 sseq1 3124 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
2018, 19anbi12d 465 . . . . . 6 (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆𝑣𝑣𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
2120rspcev 2792 . . . . 5 ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2214, 15, 17, 21syl12anc 1215 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2322ex 114 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
2410, 23impbid 128 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
254, 24bitrd 187 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ⊆ wss 3075  ∪ cuni 3743  ‘cfv 5130  Topctop 12201  intcnt 12299  neicnei 12344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-ntr 12302  df-nei 12345 This theorem is referenced by:  topssnei  12368  iscnp4  12424
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