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Mirrors > Home > MPE Home > Th. List > cnrest | Structured version Visualization version GIF version |
Description: Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnrest.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cnrest | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnrest.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | eqid 2731 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | cnf 22679 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾) |
5 | simpr 485 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
6 | 4, 5 | fssresd 6745 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾) |
7 | cnvresima 6218 | . . . 4 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝑜) = ((◡𝐹 “ 𝑜) ∩ 𝐴) | |
8 | cntop1 22673 | . . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
9 | 8 | adantr 481 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top) |
10 | 9 | adantr 481 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐾) → 𝐽 ∈ Top) |
11 | 1 | topopn 22337 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
12 | ssexg 5316 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
13 | 12 | ancoms 459 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
14 | 11, 13 | sylan 580 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
15 | 8, 14 | sylan 580 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
16 | 15 | adantr 481 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐾) → 𝐴 ∈ V) |
17 | cnima 22698 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑜 ∈ 𝐾) → (◡𝐹 “ 𝑜) ∈ 𝐽) | |
18 | 17 | adantlr 713 | . . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐾) → (◡𝐹 “ 𝑜) ∈ 𝐽) |
19 | elrestr 17356 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ (◡𝐹 “ 𝑜) ∈ 𝐽) → ((◡𝐹 “ 𝑜) ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
20 | 10, 16, 18, 19 | syl3anc 1371 | . . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐾) → ((◡𝐹 “ 𝑜) ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
21 | 7, 20 | eqeltrid 2836 | . . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐾) → (◡(𝐹 ↾ 𝐴) “ 𝑜) ∈ (𝐽 ↾t 𝐴)) |
22 | 21 | ralrimiva 3145 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → ∀𝑜 ∈ 𝐾 (◡(𝐹 ↾ 𝐴) “ 𝑜) ∈ (𝐽 ↾t 𝐴)) |
23 | 1 | toptopon 22348 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
24 | 8, 23 | sylib 217 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋)) |
25 | resttopon 22594 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
26 | 24, 25 | sylan 580 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
27 | cntop2 22674 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
28 | 27 | adantr 481 | . . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top) |
29 | 2 | toptopon 22348 | . . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
30 | 28, 29 | sylib 217 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
31 | iscn 22668 | . . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑜 ∈ 𝐾 (◡(𝐹 ↾ 𝐴) “ 𝑜) ∈ (𝐽 ↾t 𝐴)))) | |
32 | 26, 30, 31 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑜 ∈ 𝐾 (◡(𝐹 ↾ 𝐴) “ 𝑜) ∈ (𝐽 ↾t 𝐴)))) |
33 | 6, 22, 32 | mpbir2and 711 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4901 ◡ccnv 5668 ↾ cres 5671 “ cima 5672 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ↾t crest 17348 Topctop 22324 TopOnctopon 22341 Cn ccn 22657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-map 8805 df-en 8923 df-fin 8926 df-fi 9388 df-rest 17350 df-topgen 17371 df-top 22325 df-topon 22342 df-bases 22378 df-cn 22660 |
This theorem is referenced by: resthauslem 22796 imacmp 22830 connima 22858 kgencn2 22990 kgencn3 22991 xkopjcn 23089 cnmpt1res 23109 cnmpt2res 23110 qtoprest 23150 hmeores 23204 ftalem3 26506 rmulccn 32739 raddcn 32740 xrge0mulc1cn 32752 rrhre 32832 cvmliftmolem1 34103 cvmlift2lem9a 34125 cvmlift2lem9 34133 ivthALT 35024 broucube 36326 areacirclem2 36381 cnres2 36436 stoweidlem28 44517 dirkercncflem2 44593 |
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