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Theorem ssnei2 22413
Description: Any subset 𝑀 of 𝑋 containing a neighborhood 𝑁 of a set 𝑆 is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1 𝑋 = 𝐽
Assertion
Ref Expression
ssnei2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem ssnei2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simprr 771 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀𝑋)
2 neii2 22405 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
3 sstr2 3949 . . . . . . 7 (𝑔𝑁 → (𝑁𝑀𝑔𝑀))
43com12 32 . . . . . 6 (𝑁𝑀 → (𝑔𝑁𝑔𝑀))
54anim2d 612 . . . . 5 (𝑁𝑀 → ((𝑆𝑔𝑔𝑁) → (𝑆𝑔𝑔𝑀)))
65reximdv 3165 . . . 4 (𝑁𝑀 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀)))
72, 6mpan9 507 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑁𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
87adantrr 715 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
9 neips.1 . . . . 5 𝑋 = 𝐽
109neiss2 22398 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
119isnei 22400 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1210, 11syldan 591 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1312adantr 481 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
141, 8, 13mpbir2and 711 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3071  wss 3908   cuni 4863  cfv 6493  Topctop 22188  neicnei 22394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22189  df-nei 22395
This theorem is referenced by:  topssnei  22421  nllyrest  22783  nllyidm  22786  hausllycmp  22791  cldllycmp  22792  txnlly  22934  neifil  23177  utop2nei  23548  cnllycmp  24265  gneispb  42308
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