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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 5164 | . . . . . . . 8 | |
2 | funfn 5153 | . . . . . . . 8 | |
3 | funfn 5153 | . . . . . . . 8 | |
4 | 1, 2, 3 | 3imtr3i 199 | . . . . . . 7 |
5 | resss 4843 | . . . . . . . . 9 | |
6 | rnss 4769 | . . . . . . . . 9 | |
7 | 5, 6 | ax-mp 5 | . . . . . . . 8 |
8 | sstr 3105 | . . . . . . . 8 | |
9 | 7, 8 | mpan 420 | . . . . . . 7 |
10 | 4, 9 | anim12i 336 | . . . . . 6 |
11 | df-f 5127 | . . . . . 6 | |
12 | df-f 5127 | . . . . . 6 | |
13 | 10, 11, 12 | 3imtr4i 200 | . . . . 5 |
14 | 13 | a1i 9 | . . . 4 |
15 | ordelord 4303 | . . . . . . 7 | |
16 | 15 | expcom 115 | . . . . . 6 |
17 | ordin 4307 | . . . . . . 7 | |
18 | 17 | ex 114 | . . . . . 6 |
19 | 16, 18 | syli 37 | . . . . 5 |
20 | dmres 4840 | . . . . . 6 | |
21 | ordeq 4294 | . . . . . 6 | |
22 | 20, 21 | ax-mp 5 | . . . . 5 |
23 | 19, 22 | syl6ibr 161 | . . . 4 |
24 | dmss 4738 | . . . . . . . . 9 | |
25 | 5, 24 | ax-mp 5 | . . . . . . . 8 |
26 | ssralv 3161 | . . . . . . . 8 | |
27 | 25, 26 | ax-mp 5 | . . . . . . 7 |
28 | ssralv 3161 | . . . . . . . . 9 | |
29 | 25, 28 | ax-mp 5 | . . . . . . . 8 |
30 | 29 | ralimi 2495 | . . . . . . 7 |
31 | 27, 30 | syl 14 | . . . . . 6 |
32 | inss1 3296 | . . . . . . . . . . . . 13 | |
33 | 20, 32 | eqsstri 3129 | . . . . . . . . . . . 12 |
34 | simpl 108 | . . . . . . . . . . . 12 | |
35 | 33, 34 | sseldi 3095 | . . . . . . . . . . 11 |
36 | fvres 5445 | . . . . . . . . . . 11 | |
37 | 35, 36 | syl 14 | . . . . . . . . . 10 |
38 | simpr 109 | . . . . . . . . . . . 12 | |
39 | 33, 38 | sseldi 3095 | . . . . . . . . . . 11 |
40 | fvres 5445 | . . . . . . . . . . 11 | |
41 | 39, 40 | syl 14 | . . . . . . . . . 10 |
42 | 37, 41 | eleq12d 2210 | . . . . . . . . 9 |
43 | 42 | imbi2d 229 | . . . . . . . 8 |
44 | 43 | ralbidva 2433 | . . . . . . 7 |
45 | 44 | ralbiia 2449 | . . . . . 6 |
46 | 31, 45 | sylibr 133 | . . . . 5 |
47 | 46 | a1i 9 | . . . 4 |
48 | 14, 23, 47 | 3anim123d 1297 | . . 3 |
49 | df-smo 6183 | . . 3 | |
50 | df-smo 6183 | . . 3 | |
51 | 48, 49, 50 | 3imtr4g 204 | . 2 |
52 | 51 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 cin 3070 wss 3071 word 4284 con0 4285 cdm 4539 crn 4540 cres 4541 wfun 5117 wfn 5118 wf 5119 cfv 5123 wsmo 6182 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-tr 4027 df-iord 4288 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-smo 6183 |
This theorem is referenced by: smores3 6190 |
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