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Mirrors > Home > MPE Home > Th. List > ioorval | Structured version Visualization version GIF version |
Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.) |
Ref | Expression |
---|---|
ioorf.1 | β’ πΉ = (π₯ β ran (,) β¦ if(π₯ = β , β¨0, 0β©, β¨inf(π₯, β*, < ), sup(π₯, β*, < )β©)) |
Ref | Expression |
---|---|
ioorval | β’ (π΄ β ran (,) β (πΉβπ΄) = if(π΄ = β , β¨0, 0β©, β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2736 | . . 3 β’ (π₯ = π΄ β (π₯ = β β π΄ = β )) | |
2 | infeq1 9470 | . . . 4 β’ (π₯ = π΄ β inf(π₯, β*, < ) = inf(π΄, β*, < )) | |
3 | supeq1 9439 | . . . 4 β’ (π₯ = π΄ β sup(π₯, β*, < ) = sup(π΄, β*, < )) | |
4 | 2, 3 | opeq12d 4881 | . . 3 β’ (π₯ = π΄ β β¨inf(π₯, β*, < ), sup(π₯, β*, < )β© = β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©) |
5 | 1, 4 | ifbieq2d 4554 | . 2 β’ (π₯ = π΄ β if(π₯ = β , β¨0, 0β©, β¨inf(π₯, β*, < ), sup(π₯, β*, < )β©) = if(π΄ = β , β¨0, 0β©, β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©)) |
6 | ioorf.1 | . 2 β’ πΉ = (π₯ β ran (,) β¦ if(π₯ = β , β¨0, 0β©, β¨inf(π₯, β*, < ), sup(π₯, β*, < )β©)) | |
7 | opex 5464 | . . 3 β’ β¨0, 0β© β V | |
8 | opex 5464 | . . 3 β’ β¨inf(π΄, β*, < ), sup(π΄, β*, < )β© β V | |
9 | 7, 8 | ifex 4578 | . 2 β’ if(π΄ = β , β¨0, 0β©, β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©) β V |
10 | 5, 6, 9 | fvmpt 6998 | 1 β’ (π΄ β ran (,) β (πΉβπ΄) = if(π΄ = β , β¨0, 0β©, β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β c0 4322 ifcif 4528 β¨cop 4634 β¦ cmpt 5231 ran crn 5677 βcfv 6543 supcsup 9434 infcinf 9435 0cc0 11109 β*cxr 11246 < clt 11247 (,)cioo 13323 |
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-sep 5299 ax-nul 5306 ax-pr 5427 |
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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 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-iota 6495 df-fun 6545 df-fv 6551 df-sup 9436 df-inf 9437 |
This theorem is referenced by: ioorinv2 25091 ioorinv 25092 ioorcl 25093 |
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