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Mirrors > Home > MPE Home > Th. List > o1dm | Structured version Visualization version GIF version |
Description: An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
o1dm | β’ (πΉ β π(1) β dom πΉ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elo1 15470 | . . 3 β’ (πΉ β π(1) β (πΉ β (β βpm β) β§ βπ₯ β β βπ β β βπ¦ β (dom πΉ β© (π₯[,)+β))(absβ(πΉβπ¦)) β€ π)) | |
2 | 1 | simplbi 499 | . 2 β’ (πΉ β π(1) β πΉ β (β βpm β)) |
3 | cnex 11191 | . . . 4 β’ β β V | |
4 | reex 11201 | . . . 4 β’ β β V | |
5 | 3, 4 | elpm2 8868 | . . 3 β’ (πΉ β (β βpm β) β (πΉ:dom πΉβΆβ β§ dom πΉ β β)) |
6 | 5 | simprbi 498 | . 2 β’ (πΉ β (β βpm β) β dom πΉ β β) |
7 | 2, 6 | syl 17 | 1 β’ (πΉ β π(1) β dom πΉ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 βwral 3062 βwrex 3071 β© cin 3948 β wss 3949 class class class wbr 5149 dom cdm 5677 βΆwf 6540 βcfv 6544 (class class class)co 7409 βpm cpm 8821 βcc 11108 βcr 11109 +βcpnf 11245 β€ cle 11249 [,)cico 13326 abscabs 15181 π(1)co1 15430 |
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-pow 5364 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-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-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 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-pm 8823 df-o1 15434 |
This theorem is referenced by: o1bdd 15475 lo1o1 15476 o1lo1 15481 o1lo12 15482 o1co 15530 o1of2 15557 o1rlimmul 15563 o1add2 15568 o1mul2 15569 o1sub2 15570 o1dif 15574 o1cxp 26479 |
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