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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 7183 | . . 3 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
2 | df-kq 22296 | . . 3 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
3 | 1, 2 | fnmpti 6486 | . 2 ⊢ KQ Fn Top |
4 | kqt0 22348 | . . . 4 ⊢ (𝑥 ∈ Top ↔ (KQ‘𝑥) ∈ Kol2) | |
5 | 4 | biimpi 218 | . . 3 ⊢ (𝑥 ∈ Top → (KQ‘𝑥) ∈ Kol2) |
6 | 5 | rgen 3148 | . 2 ⊢ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2 |
7 | ffnfv 6877 | . 2 ⊢ (KQ:Top⟶Kol2 ↔ (KQ Fn Top ∧ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2)) | |
8 | 3, 6, 7 | mpbir2an 709 | 1 ⊢ KQ:Top⟶Kol2 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∀wral 3138 {crab 3142 ∪ cuni 4832 ↦ cmpt 5139 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 qTop cqtop 16770 Topctop 21495 Kol2ct0 21908 KQckq 22295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-qtop 16774 df-top 21496 df-topon 21513 df-t0 21915 df-kq 22296 |
This theorem is referenced by: (None) |
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