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Theorem opnneissb 13740
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 13594 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) β†’ 𝑁 βŠ† 𝑋)
32adantr 276 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 βŠ† 𝑋)
4 ssid 3177 . . . . . . 7 𝑁 βŠ† 𝑁
5 sseq2 3181 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑁))
6 sseq1 3180 . . . . . . . . 9 (𝑔 = 𝑁 β†’ (𝑔 βŠ† 𝑁 ↔ 𝑁 βŠ† 𝑁))
75, 6anbi12d 473 . . . . . . . 8 (𝑔 = 𝑁 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) ↔ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)))
87rspcev 2843 . . . . . . 7 ((𝑁 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑁 ∧ 𝑁 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
94, 8mpanr2 438 . . . . . 6 ((𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
109ad2ant2l 508 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
111isnei 13729 . . . . . 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 509 . . . . 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 944 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽) ∧ (𝑆 βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑁)) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†))
1413exp43 372 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝐽 β†’ (𝑆 βŠ† 𝑋 β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))))
15143imp 1193 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
16 ssnei 13736 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑁)
1716ex 115 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
18173ad2ant1 1018 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† 𝑁))
1915, 18impbid 129 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13582  neicnei 13723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724
This theorem is referenced by:  opnneiss  13743
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