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| Mirrors > Home > MPE Home > Th. List > qtopomap | Structured version Visualization version GIF version | ||
| Description: If 𝐹 is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| qtopomap.4 | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | 
| qtopomap.5 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | 
| qtopomap.6 | ⊢ (𝜑 → ran 𝐹 = 𝑌) | 
| qtopomap.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | 
| Ref | Expression | 
|---|---|
| qtopomap | ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qtopomap.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
| 2 | qtopomap.4 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
| 3 | qtopomap.6 | . . 3 ⊢ (𝜑 → ran 𝐹 = 𝑌) | |
| 4 | qtopss 23724 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1372 | . 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹)) | 
| 6 | cntop1 23249 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
| 7 | 1, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 8 | toptopon2 22925 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
| 9 | 7, 8 | sylib 218 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽)) | 
| 10 | cnf2 23258 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌) | |
| 11 | 9, 2, 1, 10 | syl3anc 1372 | . . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌) | 
| 12 | 11 | ffnd 6736 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽) | 
| 13 | df-fo 6566 | . . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌)) | |
| 14 | 12, 3, 13 | sylanbrc 583 | . . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌) | 
| 15 | elqtop3 23712 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
| 16 | 9, 14, 15 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))) | 
| 17 | foimacnv 6864 | . . . . . . . 8 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | |
| 18 | 14, 17 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | 
| 19 | 18 | adantrr 717 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | 
| 20 | imaeq2 6073 | . . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦))) | |
| 21 | 20 | eleq1d 2825 | . . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾)) | 
| 22 | qtopomap.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
| 23 | 22 | ralrimiva 3145 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ 𝐾) | 
| 24 | 23 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ 𝐾) | 
| 25 | simprr 772 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽) | |
| 26 | 21, 24, 25 | rspcdva 3622 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾) | 
| 27 | 19, 26 | eqeltrrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ∈ 𝐾) | 
| 28 | 27 | ex 412 | . . . 4 ⊢ (𝜑 → ((𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → 𝑦 ∈ 𝐾)) | 
| 29 | 16, 28 | sylbid 240 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾)) | 
| 30 | 29 | ssrdv 3988 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾) | 
| 31 | 5, 30 | eqssd 4000 | 1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 ∪ cuni 4906 ◡ccnv 5683 ran crn 5685 “ cima 5687 Fn wfn 6555 ⟶wf 6556 –onto→wfo 6558 ‘cfv 6560 (class class class)co 7432 qTop cqtop 17549 Topctop 22900 TopOnctopon 22917 Cn ccn 23233 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-qtop 17553 df-top 22901 df-topon 22918 df-cn 23236 | 
| This theorem is referenced by: hmeoqtop 23784 | 
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