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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 7414 | . . 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 13338 | . . . 4 β’ π:(β* Γ β*)βΆπ« β* |
4 | 0elpw 5353 | . . . 4 β’ β β π« β* | |
5 | 3, 4 | f0cli 7098 | . . 3 β’ (πββ¨π΄, π΅β©) β π« β* |
6 | 1, 5 | eqeltri 2827 | . 2 β’ (π΄ππ΅) β π« β* |
7 | ovex 7444 | . . 3 β’ (π΄ππ΅) β V | |
8 | 7 | elpw 4605 | . 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 394 = wceq 1539 β wcel 2104 {crab 3430 β wss 3947 π« cpw 4601 β¨cop 4633 class class class wbr 5147 Γ cxp 5673 βcfv 6542 (class class class)co 7411 β cmpo 7413 β*cxr 11251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-xr 11256 |
This theorem is referenced by: iccssxr 13411 iocssxr 13412 icossxr 13413 ioossioobi 44528 |
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