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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 22633 | . . . 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 6755 | . . . 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 23003 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
9 | 3, 6, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top) |
10 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) | |
11 | 7 | qtopuni 23005 | . . . . . . . 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 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
14 | 10, 13 | eleqtrrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌) |
15 | 14 | snssd 4767 | . . . 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 521 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) |
19 | istopon 22213 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
20 | 18, 19 | sylibr 233 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
21 | qtopcld 23016 | . . . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)))) |
24 | 15, 17, 23 | mpbir2and 711 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
25 | 24 | ralrimiva 3141 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
26 | eqid 2737 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
27 | 26 | ist1 22624 | . 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ⊆ wss 3908 {csn 4584 ∪ cuni 4863 ◡ccnv 5630 “ cima 5634 Fn wfn 6488 –onto→wfo 6491 ‘cfv 6493 (class class class)co 7351 qTop cqtop 17345 Topctop 22194 TopOnctopon 22211 Clsdccld 22319 Frect1 22610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-qtop 17349 df-top 22195 df-topon 22212 df-cld 22322 df-t1 22617 |
This theorem is referenced by: (None) |
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