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| Mirrors > Home > MPE Home > Th. List > ist1 | Structured version Visualization version GIF version | ||
| Description: The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.) |
| Ref | Expression |
|---|---|
| ist0.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| ist1 | ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unieq 4918 | . . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
| 2 | ist0.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
| 4 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽)) | |
| 5 | 4 | eleq2d 2827 | . . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽))) |
| 6 | 3, 5 | raleqbidv 3346 | . 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
| 7 | df-t1 23322 | . 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)} | |
| 8 | 6, 7 | elrab2 3695 | 1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 Frect1 23315 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-t1 23322 |
| This theorem is referenced by: t1sncld 23334 t1ficld 23335 t1top 23338 ist1-2 23355 cnt1 23358 ordtt1 23387 qtopt1 33834 zarmxt1 33879 onint1 36450 |
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