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Theorem neiuni 21414
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 21413 . . . 4 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
32biimpa 477 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4 elssuni 4774 . . 3 (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ((nei‘𝐽)‘𝑆))
53, 4syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ((nei‘𝐽)‘𝑆))
61neii1 21398 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
76ex 413 . . . . 5 (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
87adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
98ralrimiv 3148 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
10 unissb 4776 . . 3 ( ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
119, 10sylibr 235 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
125, 11eqssd 3906 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  wss 3859   cuni 4745  cfv 6225  Topctop 21185  neicnei 21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-nei 21390
This theorem is referenced by:  neifil  22172
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