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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 23739 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | . 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹)) |
6 | cntop1 23264 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
7 | 1, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | toptopon2 22940 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
9 | 7, 8 | sylib 218 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
10 | cnf2 23273 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌) | |
11 | 9, 2, 1, 10 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌) |
12 | 11 | ffnd 6738 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽) |
13 | df-fo 6569 | . . . . . 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 23727 | . . . . 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 6866 | . . . . . . . 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 6076 | . . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦))) | |
21 | 20 | eleq1d 2824 | . . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾)) |
22 | qtopomap.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
23 | 22 | ralrimiva 3144 | . . . . . . . 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 773 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽) | |
26 | 21, 24, 25 | rspcdva 3623 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾) |
27 | 19, 26 | eqeltrrd 2840 | . . . . 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 4001 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾) |
31 | 5, 30 | eqssd 4013 | 1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ∪ cuni 4912 ◡ccnv 5688 ran crn 5690 “ cima 5692 Fn wfn 6558 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 qTop cqtop 17550 Topctop 22915 TopOnctopon 22932 Cn ccn 23248 |
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-map 8867 df-qtop 17554 df-top 22916 df-topon 22933 df-cn 23251 |
This theorem is referenced by: hmeoqtop 23799 |
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