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Theorem opnneissb 22973
Description: An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
opnneissb ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
Proof of Theorem opnneissb
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
21eltopss 22764 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) β†’ 𝑁 βŠ† 𝑋)
32adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 βŠ† 𝑋)
4 ssid 3999 . . . . . . 7 𝑁 βŠ† 𝑁
5 sseq2 4003 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑁))
6 sseq1 4002 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑔 βŠ† 𝑁 ↔ 𝑁 βŠ† 𝑁))
75, 6anbi12d 630 . . . . . . . 8 (𝑔 = 𝑁 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)))
87rspcev 3606 . . . . . . 7 ((𝑁 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
94, 8mpanr2 701 . . . . . 6 ((𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
109ad2ant2l 743 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
111isnei 22962 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
1211ad2ant2r 744 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
133, 10, 12mpbir2and 710 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))
1413exp43 436 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝐽 β†’ (𝑆 βŠ† 𝑋 β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))))
15143imp 1108 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
16 ssnei 22969 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
1716ex 412 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
18173ad2ant1 1130 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
1915, 18impbid 211 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6537  Topctop 22750  neicnei 22956
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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 22751  df-nei 22957
This theorem is referenced by:  opnneiss  22977
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