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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 4111 | . . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽) | |
2 | unieq 4380 | . . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽) | |
3 | iscmp.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋) |
5 | 4 | eqeq1d 2612 | . . . 4 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦)) |
6 | 4 | eqeq1d 2612 | . . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧)) |
7 | 6 | rexbidv 3034 | . . . 4 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)) |
8 | 5, 7 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐽 → ((∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
9 | 1, 8 | raleqbidv 3129 | . 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 21000 | . 2 ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} | |
11 | 9, 10 | elrab2 3333 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 𝒫 cpw 4108 ∪ cuni 4372 Fincfn 7841 Topctop 20517 Compccmp 20999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-cmp 21000 |
This theorem is referenced by: cmpcov 21002 cncmp 21005 fincmp 21006 cmptop 21008 cmpsub 21013 tgcmp 21014 uncmp 21016 sscmp 21018 cmpfi 21021 comppfsc 21145 txcmp 21256 alexsubb 21660 alexsubALT 21665 cmpcref 29245 onsucsuccmpi 31612 limsucncmpi 31614 heibor 32790 |
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