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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoreelrnab | Structured version Visualization version GIF version |
Description: Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.) |
Ref | Expression |
---|---|
icoreelrnab.1 | β’ πΌ = ([,) β (β Γ β)) |
Ref | Expression |
---|---|
icoreelrnab | β’ (π β πΌ β βπ β β βπ β β π = {π§ β β β£ (π β€ π§ β§ π§ < π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoreelrnab.1 | . . . . . 6 β’ πΌ = ([,) β (β Γ β)) | |
2 | df-ima 5650 | . . . . . 6 β’ ([,) β (β Γ β)) = ran ([,) βΎ (β Γ β)) | |
3 | 1, 2 | eqtri 2761 | . . . . 5 β’ πΌ = ran ([,) βΎ (β Γ β)) |
4 | 3 | eleq2i 2826 | . . . 4 β’ (π β πΌ β π β ran ([,) βΎ (β Γ β))) |
5 | icoreresf 35873 | . . . . 5 β’ ([,) βΎ (β Γ β)):(β Γ β)βΆπ« β | |
6 | ffn 6672 | . . . . 5 β’ (([,) βΎ (β Γ β)):(β Γ β)βΆπ« β β ([,) βΎ (β Γ β)) Fn (β Γ β)) | |
7 | ovelrn 7534 | . . . . 5 β’ (([,) βΎ (β Γ β)) Fn (β Γ β) β (π β ran ([,) βΎ (β Γ β)) β βπ β β βπ β β π = (π([,) βΎ (β Γ β))π))) | |
8 | 5, 6, 7 | mp2b 10 | . . . 4 β’ (π β ran ([,) βΎ (β Γ β)) β βπ β β βπ β β π = (π([,) βΎ (β Γ β))π)) |
9 | 4, 8 | bitri 275 | . . 3 β’ (π β πΌ β βπ β β βπ β β π = (π([,) βΎ (β Γ β))π)) |
10 | ovres 7524 | . . . . 5 β’ ((π β β β§ π β β) β (π([,) βΎ (β Γ β))π) = (π[,)π)) | |
11 | 10 | eqeq2d 2744 | . . . 4 β’ ((π β β β§ π β β) β (π = (π([,) βΎ (β Γ β))π) β π = (π[,)π))) |
12 | 11 | 2rexbiia 3206 | . . 3 β’ (βπ β β βπ β β π = (π([,) βΎ (β Γ β))π) β βπ β β βπ β β π = (π[,)π)) |
13 | 9, 12 | bitri 275 | . 2 β’ (π β πΌ β βπ β β βπ β β π = (π[,)π)) |
14 | icoreval 35874 | . . . 4 β’ ((π β β β§ π β β) β (π[,)π) = {π§ β β β£ (π β€ π§ β§ π§ < π)}) | |
15 | 14 | eqeq2d 2744 | . . 3 β’ ((π β β β§ π β β) β (π = (π[,)π) β π = {π§ β β β£ (π β€ π§ β§ π§ < π)})) |
16 | 15 | 2rexbiia 3206 | . 2 β’ (βπ β β βπ β β π = (π[,)π) β βπ β β βπ β β π = {π§ β β β£ (π β€ π§ β§ π§ < π)}) |
17 | 13, 16 | bitri 275 | 1 β’ (π β πΌ β βπ β β βπ β β π = {π§ β β β£ (π β€ π§ β§ π§ < π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 {crab 3406 π« cpw 4564 class class class wbr 5109 Γ cxp 5635 ran crn 5638 βΎ cres 5639 β cima 5640 Fn wfn 6495 βΆwf 6496 (class class class)co 7361 βcr 11058 < clt 11197 β€ cle 11198 [,)cico 13275 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 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-po 5549 df-so 5550 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-1st 7925 df-2nd 7926 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-ico 13279 |
This theorem is referenced by: isbasisrelowllem1 35876 isbasisrelowllem2 35877 icoreclin 35878 |
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