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Theorem ssnei2 21659
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 769 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀𝑋)
2 neii2 21651 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
3 sstr2 3978 . . . . . . 7 (𝑔𝑁 → (𝑁𝑀𝑔𝑀))
43com12 32 . . . . . 6 (𝑁𝑀 → (𝑔𝑁𝑔𝑀))
54anim2d 611 . . . . 5 (𝑁𝑀 → ((𝑆𝑔𝑔𝑁) → (𝑆𝑔𝑔𝑀)))
65reximdv 3278 . . . 4 (𝑁𝑀 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀)))
72, 6mpan9 507 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑁𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
87adantrr 713 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
9 neips.1 . . . . 5 𝑋 = 𝐽
109neiss2 21644 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
119isnei 21646 . . . 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 709 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3144  wss 3940   cuni 4837  cfv 6354  Topctop 21436  neicnei 21640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-top 21437  df-nei 21641
This theorem is referenced by:  topssnei  21667  nllyrest  22029  nllyidm  22032  hausllycmp  22037  cldllycmp  22038  txnlly  22180  neifil  22423  utop2nei  22793  cnllycmp  23494  gneispb  40365
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