![]() |
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 12971 | . . . 4 β’ β* β V | |
2 | ssrab2 4078 | . . . 4 β’ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β β* | |
3 | 1, 2 | elpwi2 5347 | . . 3 β’ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)} β π« β* |
4 | 3 | rgen2w 3067 | . 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 8054 | . 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 397 = wceq 1542 β wcel 2107 βwral 3062 {crab 3433 Vcvv 3475 π« cpw 4603 class class class wbr 5149 Γ cxp 5675 βΆwf 6540 β cmpo 7411 β*cxr 11247 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-xr 11252 |
This theorem is referenced by: ixxex 13335 ixxssxr 13336 elixx3g 13337 ndmioo 13351 iccf 13425 iocpnfordt 22719 icomnfordt 22720 tpr2rico 32892 icoreresf 36233 icoreelrn 36242 relowlpssretop 36245 dmico 44278 |
Copyright terms: Public domain | W3C validator |