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Mirrors > Home > MPE Home > Th. List > isperf | Structured version Visualization version GIF version |
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isperf | ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6669 | . . . 4 ⊢ (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽)) | |
2 | unieq 4848 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
3 | lpfval.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | syl6eqr 2874 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
5 | 1, 4 | fveq12d 6676 | . . 3 ⊢ (𝑗 = 𝐽 → ((limPt‘𝑗)‘∪ 𝑗) = ((limPt‘𝐽)‘𝑋)) |
6 | 5, 4 | eqeq12d 2837 | . 2 ⊢ (𝑗 = 𝐽 → (((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
7 | df-perf 21744 | . 2 ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗} | |
8 | 6, 7 | elrab2 3682 | 1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∪ cuni 4837 ‘cfv 6354 Topctop 21500 limPtclp 21741 Perfcperf 21742 |
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-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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-perf 21744 |
This theorem is referenced by: isperf2 21759 perflp 21761 perftop 21763 restperf 21791 |
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