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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoreelrn | Structured version Visualization version GIF version |
Description: A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.) |
Ref | Expression |
---|---|
icoreelrn.1 | β’ πΌ = ([,) β (β Γ β)) |
Ref | Expression |
---|---|
icoreelrn | β’ ((π΄ β β β§ π΅ β β) β {π§ β β β£ (π΄ β€ π§ β§ π§ < π΅)} β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoreval 36690 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π΄[,)π΅) = {π§ β β β£ (π΄ β€ π§ β§ π§ < π΅)}) | |
2 | simpl 482 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β π΄ β β) | |
3 | simpr 484 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β π΅ β β) | |
4 | df-ico 13326 | . . . . . 6 β’ [,) = (π β β*, π β β* β¦ {π§ β β* β£ (π β€ π§ β§ π§ < π)}) | |
5 | 4 | ixxf 13330 | . . . . 5 β’ [,):(β* Γ β*)βΆπ« β* |
6 | ffun 6710 | . . . . 5 β’ ([,):(β* Γ β*)βΆπ« β* β Fun [,)) | |
7 | 5, 6 | mp1i 13 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β Fun [,)) |
8 | rexpssxrxp 11255 | . . . . . 6 β’ (β Γ β) β (β* Γ β*) | |
9 | 5 | fdmi 6719 | . . . . . 6 β’ dom [,) = (β* Γ β*) |
10 | 8, 9 | sseqtrri 4011 | . . . . 5 β’ (β Γ β) β dom [,) |
11 | 10 | a1i 11 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (β Γ β) β dom [,)) |
12 | 2, 3, 7, 11 | elovimad 7449 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄[,)π΅) β ([,) β (β Γ β))) |
13 | icoreelrn.1 | . . 3 β’ πΌ = ([,) β (β Γ β)) | |
14 | 12, 13 | eleqtrrdi 2836 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π΄[,)π΅) β πΌ) |
15 | 1, 14 | eqeltrrd 2826 | 1 β’ ((π΄ β β β§ π΅ β β) β {π§ β β β£ (π΄ β€ π§ β§ π§ < π΅)} β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 β wss 3940 π« cpw 4594 class class class wbr 5138 Γ cxp 5664 dom cdm 5666 β cima 5669 Fun wfun 6527 βΆwf 6529 (class class class)co 7401 βcr 11104 β*cxr 11243 < clt 11244 β€ cle 11245 [,)cico 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-pre-lttri 11179 ax-pre-lttrn 11180 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ico 13326 |
This theorem is referenced by: relowlssretop 36700 |
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