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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 2737 | . . 3 β’ (π₯ = π΄ β (π₯ = β β π΄ = β )) | |
2 | infeq1 9471 | . . . 4 β’ (π₯ = π΄ β inf(π₯, β*, < ) = inf(π΄, β*, < )) | |
3 | supeq1 9440 | . . . 4 β’ (π₯ = π΄ β sup(π₯, β*, < ) = sup(π΄, β*, < )) | |
4 | 2, 3 | opeq12d 4882 | . . 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 4555 | . 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 5465 | . . 3 β’ β¨0, 0β© β V | |
8 | opex 5465 | . . 3 β’ β¨inf(π΄, β*, < ), sup(π΄, β*, < )β© β V | |
9 | 7, 8 | ifex 4579 | . 2 β’ if(π΄ = β , β¨0, 0β©, β¨inf(π΄, β*, < ), sup(π΄, β*, < )β©) β V |
10 | 5, 6, 9 | fvmpt 6999 | 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 1542 β wcel 2107 β c0 4323 ifcif 4529 β¨cop 4635 β¦ cmpt 5232 ran crn 5678 βcfv 6544 supcsup 9435 infcinf 9436 0cc0 11110 β*cxr 11247 < clt 11248 (,)cioo 13324 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-sup 9437 df-inf 9438 |
This theorem is referenced by: ioorinv2 25092 ioorinv 25093 ioorcl 25094 |
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