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Mirrors > Home > MPE Home > Th. List > ixxssxr | Structured version Visualization version GIF version |
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.) |
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 |
---|---|
ixxssxr | β’ (π΄ππ΅) β β* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7412 | . . 3 β’ (π΄ππ΅) = (πββ¨π΄, π΅β©) | |
2 | ixx.1 | . . . . 5 β’ π = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)}) | |
3 | 2 | ixxf 13334 | . . . 4 β’ π:(β* Γ β*)βΆπ« β* |
4 | 0elpw 5355 | . . . 4 β’ β β π« β* | |
5 | 3, 4 | f0cli 7100 | . . 3 β’ (πββ¨π΄, π΅β©) β π« β* |
6 | 1, 5 | eqeltri 2830 | . 2 β’ (π΄ππ΅) β π« β* |
7 | ovex 7442 | . . 3 β’ (π΄ππ΅) β V | |
8 | 7 | elpw 4607 | . 2 β’ ((π΄ππ΅) β π« β* β (π΄ππ΅) β β*) |
9 | 6, 8 | mpbi 229 | 1 β’ (π΄ππ΅) β β* |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 {crab 3433 β wss 3949 π« cpw 4603 β¨cop 4635 class class class wbr 5149 Γ cxp 5675 βcfv 6544 (class class class)co 7409 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-ne 2942 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-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-xr 11252 |
This theorem is referenced by: iccssxr 13407 iocssxr 13408 icossxr 13409 ioossioobi 44230 |
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