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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 2798 | . . . . . 6 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
5 | cnextfres.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐾 | |
6 | 4, 5 | cnf 21851 | . . . . 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 21767 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
11 | 1, 8, 10 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
12 | 11 | feq2d 6473 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵)) |
13 | 7, 12 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 9, 5 | cnextfun 22669 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹)) |
15 | 1, 2, 13, 8, 14 | syl22anc 837 | . 2 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹)) |
16 | 9 | sscls 21661 | . . . . . . 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 3916 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) |
20 | 9, 5, 1, 8, 3, 18 | flfcntr 22648 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
21 | sneq 4535 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
22 | 21 | fveq2d 6649 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋})) |
23 | 22 | oveq1d 7150 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) |
24 | 23 | oveq2d 7151 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) |
25 | 24 | fveq1d 6647 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
26 | 25 | opeliunxp2 5673 | . . . . 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 21936 | . . . . . 6 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) | |
29 | 2, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
30 | 9, 5 | cnextfval 22667 | . . . . 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 837 | . . . 4 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
32 | 27, 31 | eleqtrrd 2893 | . . 3 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
33 | df-br 5031 | . . 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 6691 | . 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 1538 ∈ wcel 2111 ⊆ wss 3881 {csn 4525 〈cop 4531 ∪ cuni 4800 ∪ ciun 4881 class class class wbr 5030 × cxp 5517 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↾t crest 16686 Topctop 21498 clsccl 21623 neicnei 21702 Cn ccn 21829 Hauscha 21913 fLimf cflf 22540 CnExtccnext 22664 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-fin 8496 df-fi 8859 df-rest 16688 df-topgen 16709 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-cnext 22665 |
This theorem is referenced by: rrhqima 31365 |
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