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Theorem ssnei2 21809
 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 773 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀𝑋)
2 neii2 21801 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
3 sstr2 3900 . . . . . . 7 (𝑔𝑁 → (𝑁𝑀𝑔𝑀))
43com12 32 . . . . . 6 (𝑁𝑀 → (𝑔𝑁𝑔𝑀))
54anim2d 615 . . . . 5 (𝑁𝑀 → ((𝑆𝑔𝑔𝑁) → (𝑆𝑔𝑔𝑀)))
65reximdv 3198 . . . 4 (𝑁𝑀 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀)))
72, 6mpan9 511 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑁𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
87adantrr 717 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))
9 neips.1 . . . . 5 𝑋 = 𝐽
109neiss2 21794 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
119isnei 21796 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1210, 11syldan 595 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
1312adantr 485 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → (𝑀 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑀𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑀))))
141, 8, 13mpbir2and 713 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072   ⊆ wss 3859  ∪ cuni 4799  ‘cfv 6336  Topctop 21586  neicnei 21790 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-top 21587  df-nei 21791 This theorem is referenced by:  topssnei  21817  nllyrest  22179  nllyidm  22182  hausllycmp  22187  cldllycmp  22188  txnlly  22330  neifil  22573  utop2nei  22944  cnllycmp  23650  gneispb  41200
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