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Mirrors > Home > MPE Home > Th. List > cnextrel | Structured version Visualization version GIF version |
Description: In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.) |
Ref | Expression |
---|---|
cnextfrel.1 | ⊢ 𝐶 = ∪ 𝐽 |
cnextfrel.2 | ⊢ 𝐵 = ∪ 𝐾 |
Ref | Expression |
---|---|
cnextrel | ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5227 | . . . 4 ⊢ Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) | |
2 | 1 | rgenw 2924 | . . 3 ⊢ ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
3 | reliun 5239 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) | |
4 | 2, 3 | mpbir 221 | . 2 ⊢ Rel ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
5 | cnextfrel.1 | . . . 4 ⊢ 𝐶 = ∪ 𝐽 | |
6 | cnextfrel.2 | . . . 4 ⊢ 𝐵 = ∪ 𝐾 | |
7 | 5, 6 | cnextfval 21866 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
8 | 7 | releqd 5203 | . 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (Rel ((𝐽CnExt𝐾)‘𝐹) ↔ Rel ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
9 | 4, 8 | mpbiri 248 | 1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 ∪ cuni 4436 ∪ ciun 4520 × cxp 5112 Rel wrel 5119 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↾t crest 16081 Topctop 20698 clsccl 20822 neicnei 20901 fLimf cflf 21739 CnExtccnext 21863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 df-cnext 21864 |
This theorem is referenced by: cnextfun 21868 |
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