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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 5689 | . . . . . 6 β’ ([,) β (β Γ β)) = ran ([,) βΎ (β Γ β)) | |
3 | 1, 2 | eqtri 2760 | . . . . 5 β’ πΌ = ran ([,) βΎ (β Γ β)) |
4 | 3 | eleq2i 2825 | . . . 4 β’ (π β πΌ β π β ran ([,) βΎ (β Γ β))) |
5 | icoreresf 36228 | . . . . 5 β’ ([,) βΎ (β Γ β)):(β Γ β)βΆπ« β | |
6 | ffn 6717 | . . . . 5 β’ (([,) βΎ (β Γ β)):(β Γ β)βΆπ« β β ([,) βΎ (β Γ β)) Fn (β Γ β)) | |
7 | ovelrn 7582 | . . . . 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 274 | . . 3 β’ (π β πΌ β βπ β β βπ β β π = (π([,) βΎ (β Γ β))π)) |
10 | ovres 7572 | . . . . 5 β’ ((π β β β§ π β β) β (π([,) βΎ (β Γ β))π) = (π[,)π)) | |
11 | 10 | eqeq2d 2743 | . . . 4 β’ ((π β β β§ π β β) β (π = (π([,) βΎ (β Γ β))π) β π = (π[,)π))) |
12 | 11 | 2rexbiia 3215 | . . 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 274 | . 2 β’ (π β πΌ β βπ β β βπ β β π = (π[,)π)) |
14 | icoreval 36229 | . . . 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 2743 | . . 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 3215 | . 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 274 | 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 396 = wceq 1541 β wcel 2106 βwrex 3070 {crab 3432 π« cpw 4602 class class class wbr 5148 Γ cxp 5674 ran crn 5677 βΎ cres 5678 β cima 5679 Fn wfn 6538 βΆwf 6539 (class class class)co 7408 βcr 11108 < clt 11247 β€ cle 11248 [,)cico 13325 |
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-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 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-po 5588 df-so 5589 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-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ico 13329 |
This theorem is referenced by: isbasisrelowllem1 36231 isbasisrelowllem2 36232 icoreclin 36233 |
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