Theorem isconn 23356

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 6881	. . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
3	1, 2	ineq12d 4201	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4		unieq 4899	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
5		isconn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2789	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
7	6	preq2d 4721	. . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋})
8	3, 7	eqeq12d 2752	. 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9		df-conn 23355	. 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
10	8, 9	elrab2 3679	1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3930  ∅c0 4313  {cpr 4608  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  Clsdccld 22959  Conncconn 23354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-conn 23355

This theorem is referenced by:  isconn2  23357  connclo  23358  conndisj  23359  conntop  23360