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

Theorem iscld 20771
Description: The predicate "𝑆 is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iscld (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))

Proof of Theorem iscld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = 𝐽
21cldval 20767 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
32eleq2d 2684 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽}))
4 difeq2 3706 . . . . 5 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
54eleq1d 2683 . . . 4 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐽 ↔ (𝑋𝑆) ∈ 𝐽))
65elrab 3351 . . 3 (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽))
73, 6syl6bb 276 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
81topopn 20651 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
9 elpw2g 4797 . . . 4 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
108, 9syl 17 . . 3 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1110anbi1d 740 . 2 (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
127, 11bitrd 268 1 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2912  cdif 3557  wss 3560  𝒫 cpw 4136   cuni 4409  cfv 5857  Topctop 20638  Clsdccld 20760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-top 20639  df-cld 20763
This theorem is referenced by:  iscld2  20772  cldss  20773  cldopn  20775  topcld  20779  discld  20833  indiscld  20835  restcld  20916  ordtcld1  20941  ordtcld2  20942  hauscmp  21150  txcld  21346  ptcld  21356  qtopcld  21456  opnsubg  21851  sszcld  22560  stoweidlem57  39611
  Copyright terms: Public domain W3C validator