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

Theorem isconn 23422
Description: The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
Hypothesis
Ref Expression
isconn.1 𝑋 = 𝐽
Assertion
Ref Expression
isconn (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Proof of Theorem isconn
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
2 fveq2 6905 . . . 4 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
31, 2ineq12d 4220 . . 3 (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4 unieq 4917 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
5 isconn.1 . . . . 5 𝑋 = 𝐽
64, 5eqtr4di 2794 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
76preq2d 4739 . . 3 (𝑗 = 𝐽 → {∅, 𝑗} = {∅, 𝑋})
83, 7eqeq12d 2752 . 2 (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9 df-conn 23421 . 2 Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
108, 9elrab2 3694 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  cin 3949  c0 4332  {cpr 4627   cuni 4906  cfv 6560  Topctop 22900  Clsdccld 23025  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-conn 23421
This theorem is referenced by:  isconn2  23423  connclo  23424  conndisj  23425  conntop  23426
  Copyright terms: Public domain W3C validator