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Theorem opnneissb 23036
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 22827 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) β†’ 𝑁 βŠ† 𝑋)
32adantr 479 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 βŠ† 𝑋)
4 ssid 3995 . . . . . . 7 𝑁 βŠ† 𝑁
5 sseq2 3999 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑁))
6 sseq1 3998 . . . . . . . . 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 3601 . . . . . . 7 ((𝑁 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
94, 8mpanr2 702 . . . . . 6 ((𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
109ad2ant2l 744 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
111isnei 23025 . . . . . 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 745 . . . . 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 711 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))
1413exp43 435 . . 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 23032 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
1716ex 411 . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   βŠ† wss 3939  βˆͺ cuni 4903  β€˜cfv 6543  Topctop 22813  neicnei 23019
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22814  df-nei 23020
This theorem is referenced by:  opnneiss  23040
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