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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 2731 | . . . . . 6 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
5 | cnextfres.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐾 | |
6 | 4, 5 | cnf 23070 | . . . . 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 22986 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
11 | 1, 8, 10 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
12 | 11 | feq2d 6703 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵)) |
13 | 7, 12 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 9, 5 | cnextfun 23888 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹)) |
15 | 1, 2, 13, 8, 14 | syl22anc 836 | . 2 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹)) |
16 | 9 | sscls 22880 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
17 | 1, 8, 16 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
18 | cnextfres.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 17, 18 | sseldd 3983 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) |
20 | 9, 5, 1, 8, 3, 18 | flfcntr 23867 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
21 | sneq 4638 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
22 | 21 | fveq2d 6895 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋})) |
23 | 22 | oveq1d 7427 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) |
24 | 23 | oveq2d 7428 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) |
25 | 24 | fveq1d 6893 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
26 | 25 | opeliunxp2 5838 | . . . . 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 582 | . . . 4 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
28 | haustop 23155 | . . . . . 6 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) | |
29 | 2, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
30 | 9, 5 | cnextfval 23886 | . . . . 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 836 | . . . 4 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
32 | 27, 31 | eleqtrrd 2835 | . . 3 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
33 | df-br 5149 | . . 3 ⊢ (𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) | |
34 | 32, 33 | sylibr 233 | . 2 ⊢ (𝜑 → 𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋)) |
35 | funbrfv 6942 | . 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 1540 ∈ wcel 2105 ⊆ wss 3948 {csn 4628 〈cop 4634 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 × cxp 5674 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↾t crest 17373 Topctop 22715 clsccl 22842 neicnei 22921 Cn ccn 23048 Hauscha 23132 fLimf cflf 23759 CnExtccnext 23883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-map 8828 df-pm 8829 df-en 8946 df-fin 8949 df-fi 9412 df-rest 17375 df-topgen 17396 df-fbas 21230 df-fg 21231 df-top 22716 df-topon 22733 df-bases 22769 df-cld 22843 df-ntr 22844 df-cls 22845 df-nei 22922 df-cn 23051 df-cnp 23052 df-haus 23139 df-fil 23670 df-fm 23762 df-flim 23763 df-flf 23764 df-cnext 23884 |
This theorem is referenced by: rrhqima 33458 |
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