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Mirrors > Home > MPE Home > Th. List > iscmp | Structured version Visualization version GIF version |
Description: The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
iscmp.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscmp | ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4617 | . . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽) | |
2 | unieq 4920 | . . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
3 | iscmp.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
5 | 4 | eqeq1d 2735 | . . . 4 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦)) |
6 | 4 | eqeq1d 2735 | . . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧)) |
7 | 6 | rexbidv 3179 | . . . 4 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)) |
8 | 5, 7 | imbi12d 345 | . . 3 ⊢ (𝑥 = 𝐽 → ((∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
9 | 1, 8 | raleqbidv 3343 | . 2 ⊢ (𝑥 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
10 | df-cmp 22891 | . 2 ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} | |
11 | 9, 10 | elrab2 3687 | 1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∩ cin 3948 𝒫 cpw 4603 ∪ cuni 4909 Fincfn 8939 Topctop 22395 Compccmp 22890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 df-pw 4605 df-uni 4910 df-cmp 22891 |
This theorem is referenced by: cmpcov 22893 cncmp 22896 fincmp 22897 cmptop 22899 cmpsub 22904 tgcmp 22905 uncmp 22907 sscmp 22909 cmpfi 22912 comppfsc 23036 txcmp 23147 alexsubb 23550 alexsubALT 23555 cmpcref 32830 onsucsuccmpi 35328 limsucncmpi 35330 pibp16 36294 heibor 36689 |
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