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Theorem ssnei2 22971
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 = βˆͺ 𝐽
Assertion
Ref Expression
ssnei2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))
Proof of Theorem ssnei2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simprr 770 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 βŠ† 𝑋)
2 neii2 22963 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
3 sstr2 3984 . . . . . . 7 (𝑔 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑀 β†’ 𝑔 βŠ† 𝑀))
43com12 32 . . . . . 6 (𝑁 βŠ† 𝑀 β†’ (𝑔 βŠ† 𝑁 β†’ 𝑔 βŠ† 𝑀))
54anim2d 611 . . . . 5 (𝑁 βŠ† 𝑀 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
65reximdv 3164 . . . 4 (𝑁 βŠ† 𝑀 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
72, 6mpan9 506 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ 𝑁 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
87adantrr 714 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
9 neips.1 . . . . 5 𝑋 = βˆͺ 𝐽
109neiss2 22956 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
119isnei 22958 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
1210, 11syldan 590 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
1312adantr 480 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
141, 8, 13mpbir2and 710 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6536  Topctop 22746  neicnei 22952
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-nei 22953
This theorem is referenced by:  topssnei  22979  nllyrest  23341  nllyidm  23344  hausllycmp  23349  cldllycmp  23350  txnlly  23492  neifil  23735  utop2nei  24106  cnllycmp  24833  gneispb  43439
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