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Theorem isconn 23316
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 6897 . . . 4 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
31, 2ineq12d 4213 . . 3 (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4 unieq 4919 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
5 isconn.1 . . . . 5 𝑋 = 𝐽
64, 5eqtr4di 2786 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
76preq2d 4745 . . 3 (𝑗 = 𝐽 → {∅, 𝑗} = {∅, 𝑋})
83, 7eqeq12d 2744 . 2 (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9 df-conn 23315 . 2 Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
108, 9elrab2 3685 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  cin 3946  c0 4323  {cpr 4631   cuni 4908  cfv 6548  Topctop 22794  Clsdccld 22919  Conncconn 23314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-conn 23315
This theorem is referenced by:  isconn2  23317  connclo  23318  conndisj  23319  conntop  23320
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