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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 5655 | . . . 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 5776 | . . 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 23436 | . . 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 5738 | . 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 258 | 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 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3914 {csn 4590 βͺ cuni 4869 βͺ ciun 4958 Γ cxp 5635 Rel wrel 5642 βΆwf 6496 βcfv 6500 (class class class)co 7361 βΎt crest 17310 Topctop 22265 clsccl 22392 neicnei 22471 fLimf cflf 23309 CnExtccnext 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-pm 8774 df-cnext 23434 |
This theorem is referenced by: cnextfun 23438 |
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