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Mirrors > Home > ILE Home > Th. List > smores | Unicode version |
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
smores |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 4941 | . . . . . . . 8 | |
2 | funfn 4931 | . . . . . . . 8 | |
3 | funfn 4931 | . . . . . . . 8 | |
4 | 1, 2, 3 | 3imtr3i 189 | . . . . . . 7 |
5 | resss 4635 | . . . . . . . . 9 | |
6 | rnss 4564 | . . . . . . . . 9 | |
7 | 5, 6 | ax-mp 7 | . . . . . . . 8 |
8 | sstr 2953 | . . . . . . . 8 | |
9 | 7, 8 | mpan 400 | . . . . . . 7 |
10 | 4, 9 | anim12i 321 | . . . . . 6 |
11 | df-f 4906 | . . . . . 6 | |
12 | df-f 4906 | . . . . . 6 | |
13 | 10, 11, 12 | 3imtr4i 190 | . . . . 5 |
14 | 13 | a1i 9 | . . . 4 |
15 | ordelord 4118 | . . . . . . 7 | |
16 | 15 | expcom 109 | . . . . . 6 |
17 | ordin 4122 | . . . . . . 7 | |
18 | 17 | ex 108 | . . . . . 6 |
19 | 16, 18 | syli 33 | . . . . 5 |
20 | dmres 4632 | . . . . . 6 | |
21 | ordeq 4109 | . . . . . 6 | |
22 | 20, 21 | ax-mp 7 | . . . . 5 |
23 | 19, 22 | syl6ibr 151 | . . . 4 |
24 | dmss 4534 | . . . . . . . . 9 | |
25 | 5, 24 | ax-mp 7 | . . . . . . . 8 |
26 | ssralv 3004 | . . . . . . . 8 | |
27 | 25, 26 | ax-mp 7 | . . . . . . 7 |
28 | ssralv 3004 | . . . . . . . . 9 | |
29 | 25, 28 | ax-mp 7 | . . . . . . . 8 |
30 | 29 | ralimi 2384 | . . . . . . 7 |
31 | 27, 30 | syl 14 | . . . . . 6 |
32 | inss1 3157 | . . . . . . . . . . . . 13 | |
33 | 20, 32 | eqsstri 2975 | . . . . . . . . . . . 12 |
34 | simpl 102 | . . . . . . . . . . . 12 | |
35 | 33, 34 | sseldi 2943 | . . . . . . . . . . 11 |
36 | fvres 5198 | . . . . . . . . . . 11 | |
37 | 35, 36 | syl 14 | . . . . . . . . . 10 |
38 | simpr 103 | . . . . . . . . . . . 12 | |
39 | 33, 38 | sseldi 2943 | . . . . . . . . . . 11 |
40 | fvres 5198 | . . . . . . . . . . 11 | |
41 | 39, 40 | syl 14 | . . . . . . . . . 10 |
42 | 37, 41 | eleq12d 2108 | . . . . . . . . 9 |
43 | 42 | imbi2d 219 | . . . . . . . 8 |
44 | 43 | ralbidva 2322 | . . . . . . 7 |
45 | 44 | ralbiia 2338 | . . . . . 6 |
46 | 31, 45 | sylibr 137 | . . . . 5 |
47 | 46 | a1i 9 | . . . 4 |
48 | 14, 23, 47 | 3anim123d 1214 | . . 3 |
49 | df-smo 5901 | . . 3 | |
50 | df-smo 5901 | . . 3 | |
51 | 48, 49, 50 | 3imtr4g 194 | . 2 |
52 | 51 | impcom 116 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 cin 2916 wss 2917 word 4099 con0 4100 cdm 4345 crn 4346 cres 4347 wfun 4896 wfn 4897 wf 4898 cfv 4902 wsmo 5900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-tr 3855 df-iord 4103 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-smo 5901 |
This theorem is referenced by: smores3 5908 |
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