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Theorem neiuni 20836
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 = 𝐽
Assertion
Ref Expression
neiuni ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))

Proof of Theorem neiuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tpnei.1 . . . . 5 𝑋 = 𝐽
21tpnei 20835 . . . 4 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
32biimpa 501 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4 elssuni 4433 . . 3 (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ((nei‘𝐽)‘𝑆))
53, 4syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ((nei‘𝐽)‘𝑆))
61neii1 20820 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
76ex 450 . . . . 5 (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
87adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
98ralrimiv 2959 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
10 unissb 4435 . . 3 ( ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
119, 10sylibr 224 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
125, 11eqssd 3600 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555   cuni 4402  cfv 5847  Topctop 20617  neicnei 20811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-nei 20812
This theorem is referenced by:  neifil  21594
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