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Mirrors > Home > MPE Home > Th. List > iscon | Structured version Visualization version GIF version |
Description: The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.) |
Ref | Expression |
---|---|
iscon.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscon | ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽) | |
2 | fveq2 6103 | . . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
3 | 1, 2 | ineq12d 3777 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽))) |
4 | unieq 4380 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
5 | iscon.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
6 | 4, 5 | syl6eqr 2662 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
7 | 6 | preq2d 4219 | . . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋}) |
8 | 3, 7 | eqeq12d 2625 | . 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
9 | df-con 21025 | . 2 ⊢ Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}} | |
10 | 8, 9 | elrab2 3333 | 1 ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ∅c0 3874 {cpr 4127 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 Conccon 21024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-con 21025 |
This theorem is referenced by: iscon2 21027 conclo 21028 conndisj 21029 contop 21030 |
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