![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ixxf | Structured version Visualization version GIF version |
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
ixx.1 | β’ π = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)}) |
Ref | Expression |
---|---|
ixxf | β’ π:(β* Γ β*)βΆπ« β* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrex 12970 | . . . 4 β’ β* β V | |
2 | ssrab2 4077 | . . . 4 β’ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β β* | |
3 | 1, 2 | elpwi2 5346 | . . 3 β’ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β π« β* |
4 | 3 | rgen2w 3066 | . 2 β’ βπ₯ β β* βπ¦ β β* {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β π« β* |
5 | ixx.1 | . . 3 β’ π = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)}) | |
6 | 5 | fmpo 8053 | . 2 β’ (βπ₯ β β* βπ¦ β β* {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β π« β* β π:(β* Γ β*)βΆπ« β*) |
7 | 4, 6 | mpbi 229 | 1 β’ π:(β* Γ β*)βΆπ« β* |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 {crab 3432 Vcvv 3474 π« cpw 4602 class class class wbr 5148 Γ cxp 5674 βΆwf 6539 β cmpo 7410 β*cxr 11246 |
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-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-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-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-fv 6551 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-xr 11251 |
This theorem is referenced by: ixxex 13334 ixxssxr 13335 elixx3g 13336 ndmioo 13350 iccf 13424 iocpnfordt 22718 icomnfordt 22719 tpr2rico 32887 icoreresf 36228 icoreelrn 36237 relowlpssretop 36240 dmico 44268 |
Copyright terms: Public domain | W3C validator |