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Mirrors > Home > MPE Home > Th. List > cnextfres | Structured version Visualization version GIF version |
Description: 𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 29-Aug-2020.) |
Ref | Expression |
---|---|
cnextfres.c | ⊢ 𝐶 = ∪ 𝐽 |
cnextfres.b | ⊢ 𝐵 = ∪ 𝐾 |
cnextfres.j | ⊢ (𝜑 → 𝐽 ∈ Top) |
cnextfres.k | ⊢ (𝜑 → 𝐾 ∈ Haus) |
cnextfres.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
cnextfres.1 | ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) |
cnextfres.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
cnextfres | ⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnextfres.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
2 | cnextfres.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Haus) | |
3 | cnextfres.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) | |
4 | eqid 2738 | . . . . . 6 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
5 | cnextfres.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐾 | |
6 | 4, 5 | cnf 21990 | . . . . 5 ⊢ (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵) |
8 | cnextfres.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
9 | cnextfres.c | . . . . . . 7 ⊢ 𝐶 = ∪ 𝐽 | |
10 | 9 | restuni 21906 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
11 | 1, 8, 10 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
12 | 11 | feq2d 6484 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵)) |
13 | 7, 12 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 9, 5 | cnextfun 22808 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹)) |
15 | 1, 2, 13, 8, 14 | syl22anc 838 | . 2 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹)) |
16 | 9 | sscls 21800 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
17 | 1, 8, 16 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
18 | cnextfres.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 17, 18 | sseldd 3876 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) |
20 | 9, 5, 1, 8, 3, 18 | flfcntr 22787 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
21 | sneq 4523 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
22 | 21 | fveq2d 6672 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋})) |
23 | 22 | oveq1d 7179 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) |
24 | 23 | oveq2d 7180 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) |
25 | 24 | fveq1d 6670 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
26 | 25 | opeliunxp2 5675 | . . . . 5 ⊢ (〈𝑋, (𝐹‘𝑋)〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))) |
27 | 19, 20, 26 | sylanbrc 586 | . . . 4 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
28 | haustop 22075 | . . . . . 6 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) | |
29 | 2, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
30 | 9, 5 | cnextfval 22806 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
31 | 1, 29, 13, 8, 30 | syl22anc 838 | . . . 4 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
32 | 27, 31 | eleqtrrd 2836 | . . 3 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
33 | df-br 5028 | . . 3 ⊢ (𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) | |
34 | 32, 33 | sylibr 237 | . 2 ⊢ (𝜑 → 𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋)) |
35 | funbrfv 6714 | . 2 ⊢ (Fun ((𝐽CnExt𝐾)‘𝐹) → (𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹‘𝑋))) | |
36 | 15, 34, 35 | sylc 65 | 1 ⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ⊆ wss 3841 {csn 4513 〈cop 4519 ∪ cuni 4793 ∪ ciun 4878 class class class wbr 5027 × cxp 5517 Fun wfun 6327 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ↾t crest 16790 Topctop 21637 clsccl 21762 neicnei 21841 Cn ccn 21968 Hauscha 22052 fLimf cflf 22679 CnExtccnext 22803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-map 8432 df-pm 8433 df-en 8549 df-fin 8552 df-fi 8941 df-rest 16792 df-topgen 16813 df-fbas 20207 df-fg 20208 df-top 21638 df-topon 21655 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-cn 21971 df-cnp 21972 df-haus 22059 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-cnext 22804 |
This theorem is referenced by: rrhqima 31526 |
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