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Theorem neiint 21142
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 21141 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
323adant3 1155 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
433anibar 1421 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
5 simprrl 790 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆𝑣)
61ssntr 21096 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
763adantl2 1201 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
87adantrrl 706 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
95, 8sstrd 3819 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
109rexlimdvaa 3231 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11 simpl1 1235 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12 simpl3 1239 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁𝑋)
131ntropn 21087 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
1411, 12, 13syl2anc 575 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
15 simpr 473 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
161ntrss2 21095 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
1711, 12, 16syl2anc 575 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18 sseq2 3835 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆𝑣𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19 sseq1 3834 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
2018, 19anbi12d 618 . . . . . 6 (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆𝑣𝑣𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
2120rspcev 3513 . . . . 5 ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2214, 15, 17, 21syl12anc 856 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2322ex 399 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑆 ⊆ ((int‘𝐽)‘𝑁) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
2410, 23impbid 203 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
254, 24bitrd 270 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wrex 3108  wss 3780   cuni 4641  cfv 6110  Topctop 20931  intcnt 21055  neicnei 21135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-top 20932  df-ntr 21058  df-nei 21136
This theorem is referenced by:  opnnei  21158  topssnei  21162  iscnp4  21301  llycmpkgen2  21587  flimopn  22012  fclsneii  22054  fcfnei  22072  limcflf  23881  neiin  32669
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