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Mirrors > Home > ILE Home > Th. List > smores2 | Unicode version |
Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Ref | Expression |
---|---|
smores2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsmo2 6152 | . . . . . . 7 | |
2 | 1 | simp1bi 981 | . . . . . 6 |
3 | ffun 5245 | . . . . . 6 | |
4 | 2, 3 | syl 14 | . . . . 5 |
5 | funres 5134 | . . . . . 6 | |
6 | funfn 5123 | . . . . . 6 | |
7 | 5, 6 | sylib 121 | . . . . 5 |
8 | 4, 7 | syl 14 | . . . 4 |
9 | df-ima 4522 | . . . . . 6 | |
10 | imassrn 4862 | . . . . . 6 | |
11 | 9, 10 | eqsstrri 3100 | . . . . 5 |
12 | frn 5251 | . . . . . 6 | |
13 | 2, 12 | syl 14 | . . . . 5 |
14 | 11, 13 | sstrid 3078 | . . . 4 |
15 | df-f 5097 | . . . 4 | |
16 | 8, 14, 15 | sylanbrc 413 | . . 3 |
17 | 16 | adantr 274 | . 2 |
18 | smodm 6156 | . . 3 | |
19 | ordin 4277 | . . . . 5 | |
20 | dmres 4810 | . . . . . 6 | |
21 | ordeq 4264 | . . . . . 6 | |
22 | 20, 21 | ax-mp 5 | . . . . 5 |
23 | 19, 22 | sylibr 133 | . . . 4 |
24 | 23 | ancoms 266 | . . 3 |
25 | 18, 24 | sylan 281 | . 2 |
26 | resss 4813 | . . . . . 6 | |
27 | dmss 4708 | . . . . . 6 | |
28 | 26, 27 | ax-mp 5 | . . . . 5 |
29 | 1 | simp3bi 983 | . . . . 5 |
30 | ssralv 3131 | . . . . 5 | |
31 | 28, 29, 30 | mpsyl 65 | . . . 4 |
32 | 31 | adantr 274 | . . 3 |
33 | ordtr1 4280 | . . . . . . . . . . 11 | |
34 | 25, 33 | syl 14 | . . . . . . . . . 10 |
35 | inss1 3266 | . . . . . . . . . . . 12 | |
36 | 20, 35 | eqsstri 3099 | . . . . . . . . . . 11 |
37 | 36 | sseli 3063 | . . . . . . . . . 10 |
38 | 34, 37 | syl6 33 | . . . . . . . . 9 |
39 | 38 | expcomd 1402 | . . . . . . . 8 |
40 | 39 | imp31 254 | . . . . . . 7 |
41 | fvres 5413 | . . . . . . 7 | |
42 | 40, 41 | syl 14 | . . . . . 6 |
43 | 36 | sseli 3063 | . . . . . . . 8 |
44 | fvres 5413 | . . . . . . . 8 | |
45 | 43, 44 | syl 14 | . . . . . . 7 |
46 | 45 | ad2antlr 480 | . . . . . 6 |
47 | 42, 46 | eleq12d 2188 | . . . . 5 |
48 | 47 | ralbidva 2410 | . . . 4 |
49 | 48 | ralbidva 2410 | . . 3 |
50 | 32, 49 | mpbird 166 | . 2 |
51 | dfsmo2 6152 | . 2 | |
52 | 17, 25, 50, 51 | syl3anbrc 1150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 word 4254 con0 4255 cdm 4509 crn 4510 cres 4511 cima 4512 wfun 5087 wfn 5088 wf 5089 cfv 5093 wsmo 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-tr 3997 df-iord 4258 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-smo 6151 |
This theorem is referenced by: (None) |
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