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Theorem isconn 21389
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 6340 . . . 4 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
31, 2ineq12d 3946 . . 3 (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4 unieq 4584 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
5 isconn.1 . . . . 5 𝑋 = 𝐽
64, 5syl6eqr 2800 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
76preq2d 4407 . . 3 (𝑗 = 𝐽 → {∅, 𝑗} = {∅, 𝑋})
83, 7eqeq12d 2763 . 2 (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9 df-conn 21388 . 2 Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
108, 9elrab2 3495 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1620  wcel 2127  cin 3702  c0 4046  {cpr 4311   cuni 4576  cfv 6037  Topctop 20871  Clsdccld 20993  Conncconn 21387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-iota 6000  df-fv 6045  df-conn 21388
This theorem is referenced by:  isconn2  21390  connclo  21391  conndisj  21392  conntop  21393
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