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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 6645 | . . . 4 ⊢ (𝑗 = 𝐽 → (limPt‘𝑗) = (limPt‘𝐽)) | |
2 | unieq 4811 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
3 | lpfval.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | eqtr4di 2851 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
5 | 1, 4 | fveq12d 6652 | . . 3 ⊢ (𝑗 = 𝐽 → ((limPt‘𝑗)‘∪ 𝑗) = ((limPt‘𝐽)‘𝑋)) |
6 | 5, 4 | eqeq12d 2814 | . 2 ⊢ (𝑗 = 𝐽 → (((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗 ↔ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
7 | df-perf 21742 | . 2 ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗} | |
8 | 6, 7 | elrab2 3631 | 1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 limPtclp 21739 Perfcperf 21740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-perf 21742 |
This theorem is referenced by: isperf2 21757 perflp 21759 perftop 21761 restperf 21789 |
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