MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ntrtop Structured version   Visualization version   GIF version

Theorem ntrtop 20868
Description: The interior of a topology's underlying set is 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 20705 . 2 (𝐽 ∈ Top → 𝑋𝐽)
3 ssid 3622 . . 3 𝑋𝑋
41isopn3 20864 . . 3 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑋𝐽 ↔ ((int‘𝐽)‘𝑋) = 𝑋))
53, 4mpan2 707 . 2 (𝐽 ∈ Top → (𝑋𝐽 ↔ ((int‘𝐽)‘𝑋) = 𝑋))
62, 5mpbid 222 1 (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  wss 3572   cuni 4434  cfv 5886  Topctop 20692  intcnt 20815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-top 20693  df-ntr 20818
This theorem is referenced by:  0ntr  20869  dvidlem  23673  dveflem  23736  ioccncflimc  39867  icocncflimc  39871
  Copyright terms: Public domain W3C validator