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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 6897 | . . . 4 β’ (π = π½ β (limPtβπ) = (limPtβπ½)) | |
2 | unieq 4919 | . . . . 5 β’ (π = π½ β βͺ π = βͺ π½) | |
3 | lpfval.1 | . . . . 5 β’ π = βͺ π½ | |
4 | 2, 3 | eqtr4di 2786 | . . . 4 β’ (π = π½ β βͺ π = π) |
5 | 1, 4 | fveq12d 6904 | . . 3 β’ (π = π½ β ((limPtβπ)ββͺ π) = ((limPtβπ½)βπ)) |
6 | 5, 4 | eqeq12d 2744 | . 2 β’ (π = π½ β (((limPtβπ)ββͺ π) = βͺ π β ((limPtβπ½)βπ) = π)) |
7 | df-perf 23054 | . 2 β’ Perf = {π β Top β£ ((limPtβπ)ββͺ π) = βͺ π} | |
8 | 6, 7 | elrab2 3685 | 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 1534 β wcel 2099 βͺ cuni 4908 βcfv 6548 Topctop 22808 limPtclp 23051 Perfcperf 23052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-perf 23054 |
This theorem is referenced by: isperf2 23069 perflp 23071 perftop 23073 restperf 23101 |
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