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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 21508, 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 20731 | . . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌) | |
2 | 1 | 3ad2antl2 1224 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌) |
3 | cnima 21069 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽) | |
4 | 3 | 3ad2antl1 1223 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽) |
5 | simpl1 1064 | . . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
6 | cntop1 21044 | . . . . . . 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 | eqid 2622 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | toptopon 20722 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
10 | 7, 9 | sylib 208 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | simpl2 1065 | . . . . . . . 8 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ (TopOn‘𝑌)) | |
12 | cnf2 21053 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌) | |
13 | 10, 11, 5, 12 | syl3anc 1326 | . . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝑌) |
14 | ffn 6045 | . . . . . . 7 ⊢ (𝐹:∪ 𝐽⟶𝑌 → 𝐹 Fn ∪ 𝐽) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 Fn ∪ 𝐽) |
16 | simpl3 1066 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 = 𝑌) | |
17 | df-fo 5894 | . . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌)) | |
18 | 15, 16, 17 | sylanbrc 698 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽–onto→𝑌) |
19 | elqtop3 21506 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) | |
20 | 10, 18, 19 | syl2anc 693 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) |
21 | 2, 4, 20 | mpbir2and 957 | . . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ (𝐽 qTop 𝐹)) |
22 | 21 | ex 450 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ (𝐽 qTop 𝐹))) |
23 | 22 | ssrdv 3609 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∪ cuni 4436 ◡ccnv 5113 ran crn 5115 “ cima 5117 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 qTop cqtop 16163 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-qtop 16167 df-top 20699 df-topon 20716 df-cn 21031 |
This theorem is referenced by: qtoprest 21520 qtopomap 21521 qtopcmap 21522 |
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