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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 2737 | . . . . . 6 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴) | |
| 5 | cnextfres.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐾 | |
| 6 | 4, 5 | cnf 23254 | . . . . 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 23170 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
| 11 | 1, 8, 10 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
| 12 | 11 | feq2d 6722 | . . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∪ (𝐽 ↾t 𝐴)⟶𝐵)) |
| 13 | 7, 12 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 14 | 9, 5 | cnextfun 24072 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹)) |
| 15 | 1, 2, 13, 8, 14 | syl22anc 839 | . 2 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹)) |
| 16 | 9 | sscls 23064 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
| 17 | 1, 8, 16 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) |
| 18 | cnextfres.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
| 19 | 17, 18 | sseldd 3984 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) |
| 20 | 9, 5, 1, 8, 3, 18 | flfcntr 24051 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
| 21 | sneq 4636 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
| 22 | 21 | fveq2d 6910 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋})) |
| 23 | 22 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) |
| 24 | 23 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) |
| 25 | 24 | fveq1d 6908 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |
| 26 | 25 | opeliunxp2 5849 | . . . . 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 583 | . . . 4 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
| 28 | haustop 23339 | . . . . . 6 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) | |
| 29 | 2, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top) |
| 30 | 9, 5 | cnextfval 24070 | . . . . 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 839 | . . . 4 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
| 32 | 27, 31 | eleqtrrd 2844 | . . 3 ⊢ (𝜑 → 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
| 33 | df-br 5144 | . . 3 ⊢ (𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋) ↔ 〈𝑋, (𝐹‘𝑋)〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) | |
| 34 | 32, 33 | sylibr 234 | . 2 ⊢ (𝜑 → 𝑋((𝐽CnExt𝐾)‘𝐹)(𝐹‘𝑋)) |
| 35 | funbrfv 6957 | . 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 2108 ⊆ wss 3951 {csn 4626 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 class class class wbr 5143 × cxp 5683 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 clsccl 23026 neicnei 23105 Cn ccn 23232 Hauscha 23316 fLimf cflf 23943 CnExtccnext 24067 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-map 8868 df-pm 8869 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-haus 23323 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-cnext 24068 |
| This theorem is referenced by: rrhqima 34015 |
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