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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 6882 | . . . 4 β’ (π = π½ β (limPtβπ) = (limPtβπ½)) | |
2 | unieq 4911 | . . . . 5 β’ (π = π½ β βͺ π = βͺ π½) | |
3 | lpfval.1 | . . . . 5 β’ π = βͺ π½ | |
4 | 2, 3 | eqtr4di 2782 | . . . 4 β’ (π = π½ β βͺ π = π) |
5 | 1, 4 | fveq12d 6889 | . . 3 β’ (π = π½ β ((limPtβπ)ββͺ π) = ((limPtβπ½)βπ)) |
6 | 5, 4 | eqeq12d 2740 | . 2 β’ (π = π½ β (((limPtβπ)ββͺ π) = βͺ π β ((limPtβπ½)βπ) = π)) |
7 | df-perf 22985 | . 2 β’ Perf = {π β Top β£ ((limPtβπ)ββͺ π) = βͺ π} | |
8 | 6, 7 | elrab2 3679 | 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 395 = wceq 1533 β wcel 2098 βͺ cuni 4900 βcfv 6534 Topctop 22739 limPtclp 22982 Perfcperf 22983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-perf 22985 |
This theorem is referenced by: isperf2 23000 perflp 23002 perftop 23004 restperf 23032 |
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