Theorem isconn 22908

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.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
2		fveq2 6888	. . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
3	1, 2	ineq12d 4212	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4		unieq 4918	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
5		isconn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2790	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
7	6	preq2d 4743	. . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋})
8	3, 7	eqeq12d 2748	. 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9		df-conn 22907	. 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
10	8, 9	elrab2 3685	1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3946  ∅c0 4321  {cpr 4629  ∪ cuni 4907  ‘cfv 6540  Topctop 22386  Clsdccld 22511  Conncconn 22906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-conn 22907

This theorem is referenced by:  isconn2  22909  connclo  22910  conndisj  22911  conntop  22912