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Mirrors > Home > MPE Home > Th. List > ist0-2 | Structured version Visualization version GIF version |
Description: The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
ist0-2 | β’ (π½ β (TopOnβπ) β (π½ β Kol2 β βπ₯ β π βπ¦ β π (βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22285 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | eqid 2733 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
3 | 2 | ist0 22694 | . . . 4 β’ (π½ β Kol2 β (π½ β Top β§ βπ₯ β βͺ π½βπ¦ β βͺ π½(βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
4 | 3 | baib 537 | . . 3 β’ (π½ β Top β (π½ β Kol2 β βπ₯ β βͺ π½βπ¦ β βͺ π½(βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
5 | 1, 4 | syl 17 | . 2 β’ (π½ β (TopOnβπ) β (π½ β Kol2 β βπ₯ β βͺ π½βπ¦ β βͺ π½(βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
6 | toponuni 22286 | . . 3 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | 6 | raleqdv 3312 | . . 3 β’ (π½ β (TopOnβπ) β (βπ¦ β π (βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦) β βπ¦ β βͺ π½(βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
8 | 6, 7 | raleqbidv 3318 | . 2 β’ (π½ β (TopOnβπ) β (βπ₯ β π βπ¦ β π (βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦) β βπ₯ β βͺ π½βπ¦ β βͺ π½(βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
9 | 5, 8 | bitr4d 282 | 1 β’ (π½ β (TopOnβπ) β (π½ β Kol2 β βπ₯ β π βπ¦ β π (βπ β π½ (π₯ β π β π¦ β π) β π₯ = π¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2107 βwral 3061 βͺ cuni 4869 βcfv 6500 Topctop 22265 TopOnctopon 22282 Kol2ct0 22680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-topon 22283 df-t0 22687 |
This theorem is referenced by: ist0-3 22719 t1t0 22722 ist0-4 23103 kqt0lem 23110 tgpt0 23493 onsuct0 34966 |
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