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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 4811 | . . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
2 | ist0.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | eqtr4di 2851 | . . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
4 | fveq2 6645 | . . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽)) | |
5 | 4 | eleq2d 2875 | . . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽))) |
6 | 3, 5 | raleqbidv 3354 | . 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
7 | df-t1 21919 | . 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)} | |
8 | 6, 7 | elrab2 3631 | 1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {csn 4525 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 Clsdccld 21621 Frect1 21912 |
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 2770 |
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 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-t1 21919 |
This theorem is referenced by: t1sncld 21931 t1ficld 21932 t1top 21935 ist1-2 21952 cnt1 21955 ordtt1 21984 qtopt1 31188 zarmxt1 31233 onint1 33910 |
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