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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 6678 | . . 3 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
2 | df-kq 21497 | . . 3 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
3 | 1, 2 | fnmpti 6022 | . 2 ⊢ KQ Fn Top |
4 | kqt0 21549 | . . . 4 ⊢ (𝑥 ∈ Top ↔ (KQ‘𝑥) ∈ Kol2) | |
5 | 4 | biimpi 206 | . . 3 ⊢ (𝑥 ∈ Top → (KQ‘𝑥) ∈ Kol2) |
6 | 5 | rgen 2922 | . 2 ⊢ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2 |
7 | ffnfv 6388 | . 2 ⊢ (KQ:Top⟶Kol2 ↔ (KQ Fn Top ∧ ∀𝑥 ∈ Top (KQ‘𝑥) ∈ Kol2)) | |
8 | 3, 6, 7 | mpbir2an 955 | 1 ⊢ KQ:Top⟶Kol2 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ∀wral 2912 {crab 2916 ∪ cuni 4436 ↦ cmpt 4729 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 qTop cqtop 16163 Topctop 20698 Kol2ct0 21110 KQckq 21496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-qtop 16167 df-top 20699 df-topon 20716 df-t0 21117 df-kq 21497 |
This theorem is referenced by: (None) |
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