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Theorem neiuni 13597
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 13596 . . . 4 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
32biimpa 296 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
4 elssuni 3837 . . 3 (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑋 ((nei‘𝐽)‘𝑆))
53, 4syl 14 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ((nei‘𝐽)‘𝑆))
61neii1 13583 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥𝑋)
76ex 115 . . . . 5 (𝐽 ∈ Top → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
87adantr 276 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥𝑋))
98ralrimiv 2549 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
10 unissb 3839 . . 3 ( ((nei‘𝐽)‘𝑆) ⊆ 𝑋 ↔ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)𝑥𝑋)
119, 10sylibr 134 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
125, 11eqssd 3172 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wss 3129   cuni 3809  cfv 5216  Topctop 13433  neicnei 13574
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209
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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-top 13434  df-nei 13575
This theorem is referenced by: (None)
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