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Mirrors > Home > MPE Home > Th. List > cnss1 | Structured version Visualization version GIF version |
Description: If the topology 𝐾 is finer than 𝐽, then there are more continuous functions from 𝐾 than from 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnss1.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cnss1 | ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnss1.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | eqid 2818 | . . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿 | |
3 | 1, 2 | cnf 21782 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋⟶∪ 𝐿) |
4 | 3 | adantl 482 | . . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋⟶∪ 𝐿) |
5 | simpllr 772 | . . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐽 ⊆ 𝐾) | |
6 | cnima 21801 | . . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽) | |
7 | 6 | adantll 710 | . . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽) |
8 | 5, 7 | sseldd 3965 | . . . . 5 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐾) |
9 | 8 | ralrimiva 3179 | . . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾) |
10 | simpll 763 | . . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋)) | |
11 | cntop2 21777 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top) | |
12 | 11 | adantl 482 | . . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top) |
13 | toptopon2 21454 | . . . . . 6 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | |
14 | 12, 13 | sylib 219 | . . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿)) |
15 | iscn 21771 | . . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾))) | |
16 | 10, 14, 15 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾))) |
17 | 4, 9, 16 | mpbir2and 709 | . . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿)) |
18 | 17 | ex 413 | . 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿))) |
19 | 18 | ssrdv 3970 | 1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ∪ cuni 4830 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-top 21430 df-topon 21447 df-cn 21763 |
This theorem is referenced by: kgen2cn 22095 xkopjcn 22192 |
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