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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 2821 | . . . . . 6 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
5 | cnextfres.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐾 | |
6 | 4, 5 | cnf 21848 | . . . . 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 21764 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
11 | 1, 8, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
12 | 11 | feq2d 6495 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵)) |
13 | 7, 12 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 9, 5 | cnextfun 22666 | . . 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 21658 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
17 | 1, 8, 16 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
18 | cnextfres.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 17, 18 | sseldd 3968 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) |
20 | 9, 5, 1, 8, 3, 18 | flfcntr 22645 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
21 | sneq 4571 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
22 | 21 | fveq2d 6669 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋})) |
23 | 22 | oveq1d 7165 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) |
24 | 23 | oveq2d 7166 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) |
25 | 24 | fveq1d 6667 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
26 | 25 | opeliunxp2 5704 | . . . . 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 585 | . . . 4 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
28 | haustop 21933 | . . . . . 6 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) | |
29 | 2, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
30 | 9, 5 | cnextfval 22664 | . . . . 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 2916 | . . 3 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
33 | df-br 5060 | . . 3 ⊢ (𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) | |
34 | 32, 33 | sylibr 236 | . 2 ⊢ (𝜑 → 𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋)) |
35 | funbrfv 6711 | . 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 1533 ∈ wcel 2110 ⊆ wss 3936 {csn 4561 〈cop 4567 ∪ cuni 4832 ∪ ciun 4912 class class class wbr 5059 × cxp 5548 Fun wfun 6344 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↾t crest 16688 Topctop 21495 clsccl 21620 neicnei 21699 Cn ccn 21826 Hauscha 21910 fLimf cflf 22537 CnExtccnext 22661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-fbas 20536 df-fg 20537 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-cn 21829 df-cnp 21830 df-haus 21917 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-cnext 22662 |
This theorem is referenced by: rrhqima 31250 |
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