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Mirrors > Home > MPE Home > Th. List > qtopss | Structured version Visualization version GIF version |
Description: A surjective continuous function from 𝐽 to 𝐾 induces a topology 𝐽 qTop 𝐹 on the base set of 𝐾. This topology is in general finer than 𝐾. Together with qtopid 22602, this implies that 𝐽 qTop 𝐹 is the finest topology making 𝐹 continuous, i.e. the final topology with respect to the family {𝐹}. (Contributed by Mario Carneiro, 24-Mar-2015.) |
Ref | Expression |
---|---|
qtopss | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponss 21824 | . . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌) | |
2 | 1 | 3ad2antl2 1188 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌) |
3 | cnima 22162 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽) | |
4 | 3 | 3ad2antl1 1187 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽) |
5 | simpl1 1193 | . . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
6 | cntop1 22137 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top) |
8 | toptopon2 21815 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
9 | 7, 8 | sylib 221 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
10 | simpl2 1194 | . . . . . . . 8 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ (TopOn‘𝑌)) | |
11 | cnf2 22146 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌) | |
12 | 9, 10, 5, 11 | syl3anc 1373 | . . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝑌) |
13 | 12 | ffnd 6546 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 Fn ∪ 𝐽) |
14 | simpl3 1195 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 = 𝑌) | |
15 | df-fo 6386 | . . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌)) | |
16 | 13, 14, 15 | sylanbrc 586 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽–onto→𝑌) |
17 | elqtop3 22600 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) | |
18 | 9, 16, 17 | syl2anc 587 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) |
19 | 2, 4, 18 | mpbir2and 713 | . . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ (𝐽 qTop 𝐹)) |
20 | 19 | ex 416 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ (𝐽 qTop 𝐹))) |
21 | 20 | ssrdv 3907 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ∪ cuni 4819 ◡ccnv 5550 ran crn 5552 “ cima 5554 Fn wfn 6375 ⟶wf 6376 –onto→wfo 6378 ‘cfv 6380 (class class class)co 7213 qTop cqtop 17008 Topctop 21790 TopOnctopon 21807 Cn ccn 22121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-map 8510 df-qtop 17012 df-top 21791 df-topon 21808 df-cn 22124 |
This theorem is referenced by: qtoprest 22614 qtopomap 22615 qtopcmap 22616 |
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