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Theorem opnneissb 22609
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 22400 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) β†’ 𝑁 βŠ† 𝑋)
32adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 βŠ† 𝑋)
4 ssid 4003 . . . . . . 7 𝑁 βŠ† 𝑁
5 sseq2 4007 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑁))
6 sseq1 4006 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑔 βŠ† 𝑁 ↔ 𝑁 βŠ† 𝑁))
75, 6anbi12d 631 . . . . . . . 8 (𝑔 = 𝑁 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)))
87rspcev 3612 . . . . . . 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 22598 . . . . . 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 437 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝐽 β†’ (𝑆 βŠ† 𝑋 β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))))
15143imp 1111 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
16 ssnei 22605 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
1716ex 413 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
18173ad2ant1 1133 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6540  Topctop 22386  neicnei 22592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-nei 22593
This theorem is referenced by:  opnneiss  22613
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