Theorem isconn 22113

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 6658	. . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
3	1, 2	ineq12d 4118	. . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4		unieq 4809	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
5		isconn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	eqtr4di 2811	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
7	6	preq2d 4633	. . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋})
8	3, 7	eqeq12d 2774	. 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9		df-conn 22112	. 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
10	8, 9	elrab2 3605	1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3857  ∅c0 4225  {cpr 4524  ∪ cuni 4798  ‘cfv 6335  Topctop 21593  Clsdccld 21716  Conncconn 22111

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-conn 22112

This theorem is referenced by:  isconn2  22114  connclo  22115  conndisj  22116  conntop  22117