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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 | β’ πΆ = βͺ π½ |
cnextfrel.2 | β’ π΅ = βͺ πΎ |
Ref | Expression |
---|---|
cnextrel | β’ (((π½ β Top β§ πΎ β Top) β§ (πΉ:π΄βΆπ΅ β§ π΄ β πΆ)) β Rel ((π½CnExtπΎ)βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5694 | . . . 4 β’ Rel ({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ)) | |
2 | 1 | rgenw 3065 | . . 3 β’ βπ₯ β ((clsβπ½)βπ΄)Rel ({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ)) |
3 | reliun 5816 | . . 3 β’ (Rel βͺ π₯ β ((clsβπ½)βπ΄)({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ)) β βπ₯ β ((clsβπ½)βπ΄)Rel ({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ))) | |
4 | 2, 3 | mpbir 230 | . 2 β’ Rel βͺ π₯ β ((clsβπ½)βπ΄)({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ)) |
5 | cnextfrel.1 | . . . 4 β’ πΆ = βͺ π½ | |
6 | cnextfrel.2 | . . . 4 β’ π΅ = βͺ πΎ | |
7 | 5, 6 | cnextfval 23565 | . . 3 β’ (((π½ β Top β§ πΎ β Top) β§ (πΉ:π΄βΆπ΅ β§ π΄ β πΆ)) β ((π½CnExtπΎ)βπΉ) = βͺ π₯ β ((clsβπ½)βπ΄)({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ))) |
8 | 7 | releqd 5778 | . 2 β’ (((π½ β Top β§ πΎ β Top) β§ (πΉ:π΄βΆπ΅ β§ π΄ β πΆ)) β (Rel ((π½CnExtπΎ)βπΉ) β Rel βͺ π₯ β ((clsβπ½)βπ΄)({π₯} Γ ((πΎ fLimf (((neiβπ½)β{π₯}) βΎt π΄))βπΉ)))) |
9 | 4, 8 | mpbiri 257 | 1 β’ (((π½ β Top β§ πΎ β Top) β§ (πΉ:π΄βΆπ΅ β§ π΄ β πΆ)) β Rel ((π½CnExtπΎ)βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β wss 3948 {csn 4628 βͺ cuni 4908 βͺ ciun 4997 Γ cxp 5674 Rel wrel 5681 βΆwf 6539 βcfv 6543 (class class class)co 7408 βΎt crest 17365 Topctop 22394 clsccl 22521 neicnei 22600 fLimf cflf 23438 CnExtccnext 23562 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-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 7411 df-oprab 7412 df-mpo 7413 df-pm 8822 df-cnext 23563 |
This theorem is referenced by: cnextfun 23567 |
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