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Theorem ntrtop 21716
 Description: The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrtop (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)

Proof of Theorem ntrtop
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21topopn 21552 . 2 (𝐽 ∈ Top → 𝑋𝐽)
3 ssid 3939 . . 3 𝑋𝑋
41isopn3 21712 . . 3 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑋𝐽 ↔ ((int‘𝐽)‘𝑋) = 𝑋))
53, 4mpan2 690 . 2 (𝐽 ∈ Top → (𝑋𝐽 ↔ ((int‘𝐽)‘𝑋) = 𝑋))
62, 5mpbid 235 1 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ⊆ wss 3883  ∪ cuni 4804  ‘cfv 6332  Topctop 21539  intcnt 21663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-top 21540  df-ntr 21666 This theorem is referenced by:  0ntr  21717  dvidlem  24559  dveflem  24623  ioccncflimc  42695  icocncflimc  42699
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