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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 4426 | . . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽) | |
2 | unieq 4721 | . . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
3 | iscmp.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | syl6eqr 2832 | . . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
5 | 4 | eqeq1d 2780 | . . . 4 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦)) |
6 | 4 | eqeq1d 2780 | . . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧)) |
7 | 6 | rexbidv 3242 | . . . 4 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)) |
8 | 5, 7 | imbi12d 337 | . . 3 ⊢ (𝑥 = 𝐽 → ((∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
9 | 1, 8 | raleqbidv 3341 | . 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 21702 | . 2 ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} | |
11 | 9, 10 | elrab2 3599 | 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 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ∃wrex 3089 ∩ cin 3830 𝒫 cpw 4423 ∪ cuni 4713 Fincfn 8308 Topctop 21208 Compccmp 21701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-in 3838 df-ss 3845 df-pw 4425 df-uni 4714 df-cmp 21702 |
This theorem is referenced by: cmpcov 21704 cncmp 21707 fincmp 21708 cmptop 21710 cmpsub 21715 tgcmp 21716 uncmp 21718 sscmp 21720 cmpfi 21723 comppfsc 21847 txcmp 21958 alexsubb 22361 alexsubALT 22366 cmpcref 30758 onsucsuccmpi 33311 limsucncmpi 33313 pibp16 34135 heibor 34541 |
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