ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neiint GIF version

Theorem neiint 12096
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 12095 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
323adant3 969 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))))
433anibar 1117 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∃𝑣𝐽 (𝑆𝑣𝑣𝑁)))
5 simprrl 509 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆𝑣)
61ssntr 12073 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
763adantl2 1106 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽𝑣𝑁)) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
87adantrrl 473 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑣 ⊆ ((int‘𝐽)‘𝑁))
95, 8sstrd 3057 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ (𝑣𝐽 ∧ (𝑆𝑣𝑣𝑁))) → 𝑆 ⊆ ((int‘𝐽)‘𝑁))
109rexlimdvaa 2509 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑁) → 𝑆 ⊆ ((int‘𝐽)‘𝑁)))
11 simpl1 952 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝐽 ∈ Top)
12 simpl3 954 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → 𝑁𝑋)
131ntropn 12068 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ∈ 𝐽)
1411, 12, 13syl2anc 406 . . . . 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 12072 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝑋) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
1711, 12, 16syl2anc 406 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑁)) → ((int‘𝐽)‘𝑁) ⊆ 𝑁)
18 sseq2 3071 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑆𝑣𝑆 ⊆ ((int‘𝐽)‘𝑁)))
19 sseq1 3070 . . . . . . 7 (𝑣 = ((int‘𝐽)‘𝑁) → (𝑣𝑁 ↔ ((int‘𝐽)‘𝑁) ⊆ 𝑁))
2018, 19anbi12d 460 . . . . . 6 (𝑣 = ((int‘𝐽)‘𝑁) → ((𝑆𝑣𝑣𝑁) ↔ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)))
2120rspcev 2744 . . . . 5 ((((int‘𝐽)‘𝑁) ∈ 𝐽 ∧ (𝑆 ⊆ ((int‘𝐽)‘𝑁) ∧ ((int‘𝐽)‘𝑁) ⊆ 𝑁)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑁))
2214, 15, 17, 21syl12anc 1182 . . . 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 930   = wceq 1299  wcel 1448  wrex 2376  wss 3021   cuni 3683  cfv 5059  Topctop 11946  intcnt 12044  neicnei 12089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-top 11947  df-ntr 12047  df-nei 12090
This theorem is referenced by:  topssnei  12113  iscnp4  12168
  Copyright terms: Public domain W3C validator