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Theorem isconn 23437
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 6907 . . . 4 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
31, 2ineq12d 4229 . . 3 (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4 unieq 4923 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
5 isconn.1 . . . . 5 𝑋 = 𝐽
64, 5eqtr4di 2793 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
76preq2d 4745 . . 3 (𝑗 = 𝐽 → {∅, 𝑗} = {∅, 𝑋})
83, 7eqeq12d 2751 . 2 (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9 df-conn 23436 . 2 Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
108, 9elrab2 3698 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  c0 4339  {cpr 4633   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  Conncconn 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-conn 23436
This theorem is referenced by:  isconn2  23438  connclo  23439  conndisj  23440  conntop  23441
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