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Mirrors > Home > MPE Home > Th. List > kqf | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqf | ⊢ KQ:Top⟶Kol2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7168 | . . 3 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
2 | df-kq 22299 | . . 3 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
3 | 1, 2 | fnmpti 6463 | . 2 ⊢ KQ Fn Top |
4 | kqt0 22351 | . . . 4 ⊢ (𝑥 ∈ Top ↔ (KQ‘𝑥) ∈ Kol2) | |
5 | 4 | biimpi 219 | . . 3 ⊢ (𝑥 ∈ Top → (KQ‘𝑥) ∈ Kol2) |
6 | 5 | rgen 3116 | . 2 ⊢ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2 |
7 | ffnfv 6859 | . 2 ⊢ (KQ:Top⟶Kol2 ↔ (KQ Fn Top ∧ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2)) | |
8 | 3, 6, 7 | mpbir2an 710 | 1 ⊢ KQ:Top⟶Kol2 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∀wral 3106 {crab 3110 ∪ cuni 4800 ↦ cmpt 5110 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 qTop cqtop 16768 Topctop 21498 Kol2ct0 21911 KQckq 22298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-qtop 16772 df-top 21499 df-topon 21516 df-t0 21918 df-kq 22299 |
This theorem is referenced by: (None) |
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