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| Description: The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.) | 
| Ref | Expression | 
|---|---|
| isconn.1 | ⊢ 𝑋 = ∪ 𝐽 | 
| Ref | Expression | 
|---|---|
| isconn | ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
| 2 | fveq2 6905 | . . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
| 3 | 1, 2 | ineq12d 4220 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽))) | 
| 4 | unieq 4917 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
| 5 | isconn.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
| 6 | 4, 5 | eqtr4di 2794 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) | 
| 7 | 6 | preq2d 4739 | . . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋}) | 
| 8 | 3, 7 | eqeq12d 2752 | . 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) | 
| 9 | df-conn 23421 | . 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}} | |
| 10 | 8, 9 | elrab2 3694 | 1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ∅c0 4332 {cpr 4627 ∪ cuni 4906 ‘cfv 6560 Topctop 22900 Clsdccld 23025 Conncconn 23420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-conn 23421 | 
| This theorem is referenced by: isconn2 23423 connclo 23424 conndisj 23425 conntop 23426 | 
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