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Mirrors > Home > MPE Home > Th. List > isconn | Structured version Visualization version GIF version |
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 6663 | . . . 4 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
3 | 1, 2 | ineq12d 4187 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽))) |
4 | unieq 4838 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
5 | isconn.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
6 | 4, 5 | syl6eqr 2871 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
7 | 6 | preq2d 4668 | . . 3 ⊢ (𝑗 = 𝐽 → {∅, ∪ 𝑗} = {∅, 𝑋}) |
8 | 3, 7 | eqeq12d 2834 | . 2 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
9 | df-conn 21948 | . 2 ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}} | |
10 | 8, 9 | elrab2 3680 | 1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ∅c0 4288 {cpr 4559 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 Clsdccld 21552 Conncconn 21947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-conn 21948 |
This theorem is referenced by: isconn2 21950 connclo 21951 conndisj 21952 conntop 21953 |
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