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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 23354 | . . . 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 6823 | . . . 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 23724 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
9 | 3, 6, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top) |
10 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) | |
11 | 7 | qtopuni 23726 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
12 | 3, 4, 11 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
14 | 10, 13 | eleqtrrd 2842 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌) |
15 | 14 | snssd 4814 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌) |
16 | qtopt1.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) | |
17 | 14, 16 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) |
18 | 3, 7 | jctir 520 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) |
19 | istopon 22934 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
20 | 18, 19 | sylibr 234 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
21 | qtopcld 23737 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) | |
22 | 20, 4, 21 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) |
23 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) |
24 | 15, 17, 23 | mpbir2and 713 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
25 | 24 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
26 | eqid 2735 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
27 | 26 | ist1 23345 | . 2 ⊢ ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))) |
28 | 9, 25, 27 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 ◡ccnv 5688 “ cima 5692 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 qTop cqtop 17550 Topctop 22915 TopOnctopon 22932 Clsdccld 23040 Frect1 23331 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-qtop 17554 df-top 22916 df-topon 22933 df-cld 23043 df-t1 23338 |
This theorem is referenced by: (None) |
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