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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 22737 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | eqid 2724 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
3 | 2 | ist0 23146 | . . . 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 535 | . . 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 22738 | . . 3 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | 6 | raleqdv 3317 | . . 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 3334 | . 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 2098 βwral 3053 βͺ cuni 4899 βcfv 6533 Topctop 22717 TopOnctopon 22734 Kol2ct0 23132 |
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 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-topon 22735 df-t0 23139 |
This theorem is referenced by: ist0-3 23171 t1t0 23174 ist0-4 23555 kqt0lem 23562 tgpt0 23945 onsuct0 35816 |
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