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Theorem ssnei2 21249
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 790 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀𝑋)
2 neii2 21241 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
3 sstr2 3805 . . . . . . 7 (𝑔𝑁 → (𝑁𝑀𝑔𝑀))
43com12 32 . . . . . 6 (𝑁𝑀 → (𝑔𝑁𝑔𝑀))
54anim2d 606 . . . . 5 (𝑁𝑀 → ((𝑆𝑔𝑔𝑁) → (𝑆𝑔𝑔𝑀)))
65reximdv 3196 . . . 4 (𝑁𝑀 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀)))
72, 6mpan9 503 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑁𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
87adantrr 709 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
9 neips.1 . . . . 5 𝑋 = 𝐽
109neiss2 21234 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
119isnei 21236 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1210, 11syldan 586 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1312adantr 473 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
141, 8, 13mpbir2and 705 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  wss 3769   cuni 4628  cfv 6101  Topctop 21026  neicnei 21230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-top 21027  df-nei 21231
This theorem is referenced by:  topssnei  21257  nllyrest  21618  nllyidm  21621  hausllycmp  21626  cldllycmp  21627  txnlly  21769  neifil  22012  utop2nei  22382  cnllycmp  23083  gneispb  39211
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