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Mirrors > Home > MPE Home > Th. List > lo1f | Structured version Visualization version GIF version |
Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1f | β’ (πΉ β β€π(1) β πΉ:dom πΉβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ello1 15461 | . . 3 β’ (πΉ β β€π(1) β (πΉ β (β βpm β) β§ βπ₯ β β βπ β β βπ¦ β (dom πΉ β© (π₯[,)+β))(πΉβπ¦) β€ π)) | |
2 | 1 | simplbi 497 | . 2 β’ (πΉ β β€π(1) β πΉ β (β βpm β)) |
3 | reex 11198 | . . . 4 β’ β β V | |
4 | 3, 3 | elpm2 8865 | . . 3 β’ (πΉ β (β βpm β) β (πΉ:dom πΉβΆβ β§ dom πΉ β β)) |
5 | 4 | simplbi 497 | . 2 β’ (πΉ β (β βpm β) β πΉ:dom πΉβΆβ) |
6 | 2, 5 | syl 17 | 1 β’ (πΉ β β€π(1) β πΉ:dom πΉβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 βwral 3053 βwrex 3062 β© cin 3940 β wss 3941 class class class wbr 5139 dom cdm 5667 βΆwf 6530 βcfv 6534 (class class class)co 7402 βpm cpm 8818 βcr 11106 +βcpnf 11244 β€ cle 11248 [,)cico 13327 β€π(1)clo1 15433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-pm 8820 df-lo1 15437 |
This theorem is referenced by: lo1res 15505 lo1mptrcl 15568 |
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