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Theorem neiuni 22401
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 22400 . . . 4 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
32biimpa 478 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4 elssuni 4897 . . 3 (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ((nei‘𝐽)‘𝑆))
53, 4syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ((nei‘𝐽)‘𝑆))
61neii1 22385 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
76ex 414 . . . . 5 (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
87adantr 482 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
98ralrimiv 3141 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
10 unissb 4899 . . 3 ( ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
119, 10sylibr 233 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
125, 11eqssd 3960 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  wss 3909   cuni 4864  cfv 6492  Topctop 22170  neicnei 22376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22171  df-nei 22377
This theorem is referenced by:  neifil  23159
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