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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 4839 | . . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
2 | ist0.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
4 | fveq2 6664 | . . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽)) | |
5 | 4 | eleq2d 2898 | . . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽))) |
6 | 3, 5 | raleqbidv 3401 | . 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
7 | df-t1 21916 | . 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)} | |
8 | 6, 7 | elrab2 3682 | 1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {csn 4560 ∪ cuni 4831 ‘cfv 6349 Topctop 21495 Clsdccld 21618 Frect1 21909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-t1 21916 |
This theorem is referenced by: t1sncld 21928 t1ficld 21929 t1top 21932 ist1-2 21949 cnt1 21952 ordtt1 21981 qtopt1 31094 onint1 33792 |
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