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 21866 | . . . 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 6585 | . . . 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 22236 | . . 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 22238 | . . . . . . . 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 2913 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌) |
15 | 14 | snssd 4734 | . . . 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 21448 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
20 | 18, 19 | sylibr 235 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
21 | qtopcld 22249 | . . . . . 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 709 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
25 | 24 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
26 | eqid 2818 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
27 | 26 | ist1 21857 | . 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 {csn 4557 ∪ cuni 4830 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 qTop cqtop 16764 Topctop 21429 TopOnctopon 21446 Clsdccld 21552 Frect1 21843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-qtop 16768 df-top 21430 df-topon 21447 df-cld 21555 df-t1 21850 |
This theorem is referenced by: (None) |
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