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Mirrors > Home > ILE Home > Th. List > ioof | Unicode version |
Description: The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
ioof |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval 9684 | . . . 4 | |
2 | ioossre 9711 | . . . . 5 | |
3 | df-ov 5770 | . . . . . . 7 | |
4 | iooex 9683 | . . . . . . . 8 | |
5 | vex 2684 | . . . . . . . . 9 | |
6 | vex 2684 | . . . . . . . . 9 | |
7 | 5, 6 | opex 4146 | . . . . . . . 8 |
8 | 4, 7 | fvex 5434 | . . . . . . 7 |
9 | 3, 8 | eqeltri 2210 | . . . . . 6 |
10 | 9 | elpw 3511 | . . . . 5 |
11 | 2, 10 | mpbir 145 | . . . 4 |
12 | 1, 11 | eqeltrrdi 2229 | . . 3 |
13 | 12 | rgen2a 2484 | . 2 |
14 | df-ioo 9668 | . . 3 | |
15 | 14 | fmpo 6092 | . 2 |
16 | 13, 15 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wcel 1480 wral 2414 crab 2418 cvv 2681 wss 3066 cpw 3505 cop 3525 class class class wbr 3924 cxp 4532 wf 5114 cfv 5118 (class class class)co 5767 cr 7612 cxr 7792 clt 7793 cioo 9664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-ioo 9668 |
This theorem is referenced by: unirnioo 9749 dfioo2 9750 ioorebasg 9751 qtopbasss 12679 retopbas 12681 tgioo 12704 tgqioo 12705 |
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