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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qtopt1 | Structured version Visualization version GIF version |
Description: If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.) |
Ref | Expression |
---|---|
qtopt1.x | ⊢ 𝑋 = ∪ 𝐽 |
qtopt1.1 | ⊢ (𝜑 → 𝐽 ∈ Fre) |
qtopt1.2 | ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
qtopt1.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) |
Ref | Expression |
---|---|
qtopt1 | ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qtopt1.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Fre) | |
2 | t1top 21642 | . . . 4 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
4 | qtopt1.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) | |
5 | fofn 6421 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
7 | qtopt1.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
8 | 7 | qtoptop 22012 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
9 | 3, 6, 8 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top) |
10 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) | |
11 | 7 | qtopuni 22014 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
12 | 3, 4, 11 | syl2anc 576 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
13 | 12 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
14 | 10, 13 | eleqtrrd 2870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌) |
15 | 14 | snssd 4616 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌) |
16 | qtopt1.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) | |
17 | 14, 16 | syldan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) |
18 | 3, 7 | jctir 513 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) |
19 | istopon 21224 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
20 | 18, 19 | sylibr 226 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
21 | qtopcld 22025 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) | |
22 | 20, 4, 21 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) |
23 | 22 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) |
24 | 15, 17, 23 | mpbir2and 700 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
25 | 24 | ralrimiva 3133 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
26 | eqid 2779 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
27 | 26 | ist1 21633 | . 2 ⊢ ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))) |
28 | 9, 25, 27 | sylanbrc 575 | 1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3089 ⊆ wss 3830 {csn 4441 ∪ cuni 4712 ◡ccnv 5406 “ cima 5410 Fn wfn 6183 –onto→wfo 6186 ‘cfv 6188 (class class class)co 6976 qTop cqtop 16632 Topctop 21205 TopOnctopon 21222 Clsdccld 21328 Frect1 21619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-qtop 16636 df-top 21206 df-topon 21223 df-cld 21331 df-t1 21626 |
This theorem is referenced by: (None) |
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