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Theorem neiuni 22848
Description: The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
tpnei.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neiuni ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
Proof of Theorem neiuni
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tpnei.1 . . . . 5 𝑋 = βˆͺ 𝐽
21tpnei 22847 . . . 4 (𝐽 ∈ Top β†’ (𝑆 βŠ† 𝑋 ↔ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†)))
32biimpa 475 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 ∈ ((neiβ€˜π½)β€˜π‘†))
4 elssuni 4942 . . 3 (𝑋 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑋 βŠ† βˆͺ ((neiβ€˜π½)β€˜π‘†))
53, 4syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 βŠ† βˆͺ ((neiβ€˜π½)β€˜π‘†))
61neii1 22832 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ π‘₯ βŠ† 𝑋)
76ex 411 . . . . 5 (𝐽 ∈ Top β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) β†’ π‘₯ βŠ† 𝑋))
87adantr 479 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†) β†’ π‘₯ βŠ† 𝑋))
98ralrimiv 3143 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)π‘₯ βŠ† 𝑋)
10 unissb 4944 . . 3 (βˆͺ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝑋 ↔ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)π‘₯ βŠ† 𝑋)
119, 10sylibr 233 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝑋)
125, 11eqssd 4000 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑋 = βˆͺ ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22617  neicnei 22823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22618  df-nei 22824
This theorem is referenced by:  neifil  23606
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