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Mirrors > Home > MPE Home > Th. List > iscnrm2 | Structured version Visualization version GIF version |
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
iscnrm2 | β’ (π½ β (TopOnβπ) β (π½ β CNrm β βπ₯ β π« π(π½ βΎt π₯) β Nrm)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22759 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | eqid 2724 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
3 | 2 | iscnrm 23171 | . . . 4 β’ (π½ β CNrm β (π½ β Top β§ βπ₯ β π« βͺ π½(π½ βΎt π₯) β Nrm)) |
4 | 3 | baib 535 | . . 3 β’ (π½ β Top β (π½ β CNrm β βπ₯ β π« βͺ π½(π½ βΎt π₯) β Nrm)) |
5 | 1, 4 | syl 17 | . 2 β’ (π½ β (TopOnβπ) β (π½ β CNrm β βπ₯ β π« βͺ π½(π½ βΎt π₯) β Nrm)) |
6 | toponuni 22760 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | 6 | pweqd 4612 | . . 3 β’ (π½ β (TopOnβπ) β π« π = π« βͺ π½) |
8 | 7 | raleqdv 3317 | . 2 β’ (π½ β (TopOnβπ) β (βπ₯ β π« π(π½ βΎt π₯) β Nrm β βπ₯ β π« βͺ π½(π½ βΎt π₯) β Nrm)) |
9 | 5, 8 | bitr4d 282 | 1 β’ (π½ β (TopOnβπ) β (π½ β CNrm β βπ₯ β π« π(π½ βΎt π₯) β Nrm)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2098 βwral 3053 π« cpw 4595 βͺ cuni 4900 βcfv 6534 (class class class)co 7402 βΎt crest 17371 Topctop 22739 TopOnctopon 22756 Nrmcnrm 23158 CNrmccnrm 23159 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 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-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-topon 22757 df-cnrm 23166 |
This theorem is referenced by: restcnrm 23210 |
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