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Mirrors > Home > ILE Home > Th. List > cnss2 | 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 2164 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | cnss2.1 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
3 | 1, 2 | cnf 12745 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶𝑌) |
4 | 3 | adantl 275 | . . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶𝑌) |
5 | simplr 520 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ⊆ 𝐾) | |
6 | cnima 12761 | . . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝑓 “ 𝑥) ∈ 𝐽) | |
7 | 6 | ralrimiva 2537 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽) |
8 | 7 | adantl 275 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽) |
9 | ssralv 3201 | . . . . 5 ⊢ (𝐿 ⊆ 𝐾 → (∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽 → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)) | |
10 | 5, 8, 9 | sylc 62 | . . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽) |
11 | cntop1 12742 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | 11 | adantl 275 | . . . . . 6 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top) |
13 | 1 | toptopon 12557 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
14 | 12, 13 | sylib 121 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
15 | simpll 519 | . . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌)) | |
16 | iscn 12738 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽))) | |
17 | 14, 15, 16 | syl2anc 409 | . . . 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 933 | . . 3 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿)) |
19 | 18 | ex 114 | . 2 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿))) |
20 | 19 | ssrdv 3143 | 1 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ⊆ wss 3111 ∪ cuni 3783 ◡ccnv 4597 “ cima 4601 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 Topctop 12536 TopOnctopon 12549 Cn ccn 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-top 12537 df-topon 12550 df-cn 12729 |
This theorem is referenced by: (None) |
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