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Theorem neiint 21396
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 21395 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
323adant3 1125 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
433anibar 1322 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
5 simprrl 777 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆𝑣)
61ssntr 21350 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
763adantl2 1160 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
87adantrrl 720 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
95, 8sstrd 3899 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
109rexlimdvaa 3248 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11 simpl1 1184 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12 simpl3 1186 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁𝑋)
131ntropn 21341 . . . . . 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 485 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
161ntrss2 21349 . . . . . 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 3914 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆𝑣𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19 sseq1 3913 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
2018, 19anbi12d 630 . . . . . 6 (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆𝑣𝑣𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
2120rspcev 3559 . . . . 5 ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2214, 15, 17, 21syl12anc 833 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2322ex 413 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
2410, 23impbid 213 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
254, 24bitrd 280 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  wss 3859   cuni 4745  cfv 6225  Topctop 21185  intcnt 21309  neicnei 21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-ntr 21312  df-nei 21390
This theorem is referenced by:  opnnei  21412  topssnei  21416  iscnp4  21555  llycmpkgen2  21842  flimopn  22267  fclsneii  22309  fcfnei  22327  limcflf  24162  neiin  33289
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