Theorem isconn 23453

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 6863	. . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
3	1, 2	ineq12d 4173	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4		unieq 4875	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
5		isconn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2814	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
7	6	preq2d 4698	. . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋})
8	3, 7	eqeq12d 2777	. 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9		df-conn 23452	. 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
10	8, 9	elrab2 3653	1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3903  ∅c0 4285  {cpr 4583  ∪ cuni 4864  ‘cfv 6517  Topctop 22933  Clsdccld 23056  Conncconn 23451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-conn 23452

This theorem is referenced by:  isconn2  23454  connclo  23455  conndisj  23456  conntop  23457