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Theorem neiint 23127
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 23126 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
323adant3 1131 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
433anibar 1328 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
5 simprrl 781 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆𝑣)
61ssntr 23081 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
763adantl2 1166 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
87adantrrl 724 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
95, 8sstrd 4005 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
109rexlimdvaa 3153 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11 simpl1 1190 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12 simpl3 1192 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁𝑋)
131ntropn 23072 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
1411, 12, 13syl2anc 584 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15 simpr 484 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
161ntrss2 23080 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
1711, 12, 16syl2anc 584 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18 sseq2 4021 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆𝑣𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19 sseq1 4020 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
2018, 19anbi12d 632 . . . . . 6 (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆𝑣𝑣𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
2120rspcev 3621 . . . . 5 ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2214, 15, 17, 21syl12anc 837 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2322ex 412 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
2410, 23impbid 212 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
254, 24bitrd 279 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wrex 3067  wss 3962   cuni 4911  cfv 6562  Topctop 22914  intcnt 23040  neicnei 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-ntr 23043  df-nei 23121
This theorem is referenced by:  opnnei  23143  topssnei  23147  iscnp4  23286  llycmpkgen2  23573  flimopn  23998  fclsneii  24040  fcfnei  24058  limcflf  25930  neiin  36314  cnneiima  48712
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