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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 23606 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹)) |
6 | cntop1 23131 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
7 | 1, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | toptopon2 22807 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
9 | 7, 8 | sylib 217 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
10 | cnf2 23140 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌) | |
11 | 9, 2, 1, 10 | syl3anc 1369 | . . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌) |
12 | 11 | ffnd 6717 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽) |
13 | df-fo 6548 | . . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌)) | |
14 | 12, 3, 13 | sylanbrc 582 | . . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌) |
15 | elqtop3 23594 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
16 | 9, 14, 15 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))) |
17 | foimacnv 6850 | . . . . . . . 8 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) | |
18 | 14, 17 | sylan 579 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) |
19 | 18 | adantrr 716 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦) |
20 | imaeq2 6053 | . . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦))) | |
21 | 20 | eleq1d 2813 | . . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾)) |
22 | qtopomap.7 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | |
23 | 22 | ralrimiva 3141 | . . . . . . . 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 3608 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝐾) |
27 | 19, 26 | eqeltrrd 2829 | . . . . 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 239 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾)) |
30 | 29 | ssrdv 3984 | . 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾) |
31 | 5, 30 | eqssd 3995 | 1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 ∪ cuni 4903 ◡ccnv 5671 ran crn 5673 “ cima 5675 Fn wfn 6537 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 qTop cqtop 17476 Topctop 22782 TopOnctopon 22799 Cn ccn 23115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8838 df-qtop 17480 df-top 22783 df-topon 22800 df-cn 23118 |
This theorem is referenced by: hmeoqtop 23666 |
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