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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 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
isperf | β’ (π½ β Perf β (π½ β Top β§ ((limPtβπ½)βπ) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . 4 β’ (π = π½ β (limPtβπ) = (limPtβπ½)) | |
2 | unieq 4874 | . . . . 5 β’ (π = π½ β βͺ π = βͺ π½) | |
3 | lpfval.1 | . . . . 5 β’ π = βͺ π½ | |
4 | 2, 3 | eqtr4di 2794 | . . . 4 β’ (π = π½ β βͺ π = π) |
5 | 1, 4 | fveq12d 6846 | . . 3 β’ (π = π½ β ((limPtβπ)ββͺ π) = ((limPtβπ½)βπ)) |
6 | 5, 4 | eqeq12d 2752 | . 2 β’ (π = π½ β (((limPtβπ)ββͺ π) = βͺ π β ((limPtβπ½)βπ) = π)) |
7 | df-perf 22472 | . 2 β’ Perf = {π β Top β£ ((limPtβπ)ββͺ π) = βͺ π} | |
8 | 6, 7 | elrab2 3646 | 1 β’ (π½ β Perf β (π½ β Top β§ ((limPtβπ½)βπ) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βͺ cuni 4863 βcfv 6493 Topctop 22226 limPtclp 22469 Perfcperf 22470 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-perf 22472 |
This theorem is referenced by: isperf2 22487 perflp 22489 perftop 22491 restperf 22519 |
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