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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 23338 | . . . 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 6822 | . . . 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 23708 | . . 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 23710 | . . . . . . . 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 2844 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌) |
| 15 | 14 | snssd 4809 | . . . 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 22918 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
| 20 | 18, 19 | sylibr 234 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| 21 | qtopcld 23721 | . . . . . 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 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))) |
| 26 | eqid 2737 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
| 27 | 26 | ist1 23329 | . 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 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 qTop cqtop 17548 Topctop 22899 TopOnctopon 22916 Clsdccld 23024 Frect1 23315 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-qtop 17552 df-top 22900 df-topon 22917 df-cld 23027 df-t1 23322 |
| This theorem is referenced by: (None) |
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