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Mirrors > Home > MPE Home > Th. List > kqtopon | 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 |
---|---|
kqval.2 | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) |
Ref | Expression |
---|---|
kqtopon | ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kqval.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
2 | 1 | kqval 23713 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹)) |
3 | 1 | kqffn 23712 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) |
4 | dffn4 6820 | . . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋–onto→ran 𝐹) |
6 | qtoptopon 23691 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) | |
7 | 5, 6 | mpdan 685 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) |
8 | 2, 7 | eqeltrd 2825 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3418 ↦ cmpt 5235 ran crn 5682 Fn wfn 6548 –onto→wfo 6551 ‘cfv 6553 (class class class)co 7423 qTop cqtop 17513 TopOnctopon 22895 KQckq 23680 |
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 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-qtop 17517 df-top 22879 df-topon 22896 df-kq 23681 |
This theorem is referenced by: kqt0lem 23723 isr0 23724 r0cld 23725 regr1lem2 23727 kqreglem1 23728 kqreglem2 23729 kqnrmlem1 23730 kqnrmlem2 23731 kqtop 23732 |
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