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Mirrors > Home > MPE Home > Th. List > cnss2 | Structured version Visualization version GIF version |
Description: If the topology 𝐾 is finer than 𝐽, then there are fewer continuous functions into 𝐾 than into 𝐽 from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnss2.1 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
cnss2 | ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | cnss2.1 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
3 | 1, 2 | cnf 21848 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶𝑌) |
4 | 3 | adantl 484 | . . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶𝑌) |
5 | simplr 767 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ⊆ 𝐾) | |
6 | cnima 21867 | . . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝑓 “ 𝑥) ∈ 𝐽) | |
7 | 6 | ralrimiva 3182 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽) |
8 | 7 | adantl 484 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽) |
9 | ssralv 4032 | . . . . 5 ⊢ (𝐿 ⊆ 𝐾 → (∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽 → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)) | |
10 | 5, 8, 9 | sylc 65 | . . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽) |
11 | cntop1 21842 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 11 | adantl 484 | . . . . . 6 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top) |
13 | toptopon2 21520 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
14 | 12, 13 | sylib 220 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
15 | simpll 765 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌)) | |
16 | iscn 21837 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽))) | |
17 | 14, 15, 16 | syl2anc 586 | . . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽))) |
18 | 4, 10, 17 | mpbir2and 711 | . . 3 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿)) |
19 | 18 | ex 415 | . 2 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿))) |
20 | 19 | ssrdv 3972 | 1 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∪ cuni 4831 ◡ccnv 5548 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Topctop 21495 TopOnctopon 21512 Cn ccn 21826 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-top 21496 df-topon 21513 df-cn 21829 |
This theorem is referenced by: kgencn3 22160 xmetdcn 23440 |
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