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Mirrors > Home > MPE Home > Th. List > lo1resb | Structured version Visualization version Unicode version |
Description: The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1resb.1 | |
lo1resb.2 | |
lo1resb.3 |
Ref | Expression |
---|---|
lo1resb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lo1res 14290 | . 2 | |
2 | lo1resb.1 | . . . . . . 7 | |
3 | 2 | feqmptd 6249 | . . . . . 6 |
4 | 3 | reseq1d 5395 | . . . . 5 |
5 | resmpt3 5450 | . . . . 5 | |
6 | 4, 5 | syl6eq 2672 | . . . 4 |
7 | 6 | eleq1d 2686 | . . 3 |
8 | inss1 3833 | . . . . . 6 | |
9 | lo1resb.2 | . . . . . 6 | |
10 | 8, 9 | syl5ss 3614 | . . . . 5 |
11 | 8 | sseli 3599 | . . . . . 6 |
12 | ffvelrn 6357 | . . . . . 6 | |
13 | 2, 11, 12 | syl2an 494 | . . . . 5 |
14 | 10, 13 | ello1mpt 14252 | . . . 4 |
15 | elin 3796 | . . . . . . . . . 10 | |
16 | 15 | imbi1i 339 | . . . . . . . . 9 |
17 | impexp 462 | . . . . . . . . 9 | |
18 | 16, 17 | bitri 264 | . . . . . . . 8 |
19 | impexp 462 | . . . . . . . . . 10 | |
20 | lo1resb.3 | . . . . . . . . . . . . . . 15 | |
21 | 20 | ad2antrr 762 | . . . . . . . . . . . . . 14 |
22 | 9 | adantr 481 | . . . . . . . . . . . . . . 15 |
23 | 22 | sselda 3603 | . . . . . . . . . . . . . 14 |
24 | elicopnf 12269 | . . . . . . . . . . . . . . 15 | |
25 | 24 | baibd 948 | . . . . . . . . . . . . . 14 |
26 | 21, 23, 25 | syl2anc 693 | . . . . . . . . . . . . 13 |
27 | 26 | anbi1d 741 | . . . . . . . . . . . 12 |
28 | simplrl 800 | . . . . . . . . . . . . 13 | |
29 | maxle 12022 | . . . . . . . . . . . . 13 | |
30 | 21, 28, 23, 29 | syl3anc 1326 | . . . . . . . . . . . 12 |
31 | 27, 30 | bitr4d 271 | . . . . . . . . . . 11 |
32 | 31 | imbi1d 331 | . . . . . . . . . 10 |
33 | 19, 32 | syl5bbr 274 | . . . . . . . . 9 |
34 | 33 | pm5.74da 723 | . . . . . . . 8 |
35 | 18, 34 | syl5bb 272 | . . . . . . 7 |
36 | 35 | ralbidv2 2984 | . . . . . 6 |
37 | 2 | adantr 481 | . . . . . . 7 |
38 | simprl 794 | . . . . . . . 8 | |
39 | 20 | adantr 481 | . . . . . . . 8 |
40 | 38, 39 | ifcld 4131 | . . . . . . 7 |
41 | simprr 796 | . . . . . . 7 | |
42 | ello12r 14248 | . . . . . . . 8 | |
43 | 42 | 3expia 1267 | . . . . . . 7 |
44 | 37, 22, 40, 41, 43 | syl22anc 1327 | . . . . . 6 |
45 | 36, 44 | sylbid 230 | . . . . 5 |
46 | 45 | rexlimdvva 3038 | . . . 4 |
47 | 14, 46 | sylbid 230 | . . 3 |
48 | 7, 47 | sylbid 230 | . 2 |
49 | 1, 48 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 wral 2912 wrex 2913 cin 3573 wss 3574 cif 4086 class class class wbr 4653 cmpt 4729 cres 5116 wf 5884 cfv 5888 (class class class)co 6650 cr 9935 cpnf 10071 cle 10075 cico 12177 clo1 14218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ico 12181 df-lo1 14222 |
This theorem is referenced by: lo1eq 14299 |
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