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Theorem smores 5848
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.)
Assertion
Ref Expression
smores  Smo  dom 
Smo  |`

Proof of Theorem smores
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funres 4884 . . . . . . . 8  Fun  Fun  |`
2 funfn 4874 . . . . . . . 8  Fun  Fn  dom
3 funfn 4874 . . . . . . . 8  Fun  |`  |`  Fn  dom  |`
41, 2, 33imtr3i 189 . . . . . . 7  Fn  dom  |`  Fn  dom  |`
5 resss 4578 . . . . . . . . 9  |`  C_
6 rnss 4507 . . . . . . . . 9  |`  C_  ran  |`  C_  ran
75, 6ax-mp 7 . . . . . . . 8  ran  |`  C_  ran
8 sstr 2947 . . . . . . . 8  ran  |` 
C_  ran  ran  C_  On  ran  |` 
C_  On
97, 8mpan 400 . . . . . . 7  ran  C_  On  ran  |` 
C_  On
104, 9anim12i 321 . . . . . 6  Fn  dom  ran  C_  On  |`  Fn  dom  |`  ran  |` 
C_  On
11 df-f 4849 . . . . . 6  : dom  --> On  Fn  dom  ran  C_  On
12 df-f 4849 . . . . . 6  |`  : dom  |`  --> On  |`  Fn  dom  |`  ran  |` 
C_  On
1310, 11, 123imtr4i 190 . . . . 5  : dom  --> On  |`  : dom  |`  --> On
1413a1i 9 . . . 4  dom  : dom  --> On  |`  : dom  |`  --> On
15 ordelord 4084 . . . . . . 7  Ord  dom  dom  Ord
1615expcom 109 . . . . . 6  dom  Ord  dom  Ord
17 ordin 4088 . . . . . . 7  Ord  Ord  dom  Ord  i^i  dom
1817ex 108 . . . . . 6  Ord  Ord  dom  Ord  i^i  dom
1916, 18syli 33 . . . . 5  dom  Ord  dom  Ord  i^i  dom
20 dmres 4575 . . . . . 6  dom  |`  i^i  dom
21 ordeq 4075 . . . . . 6  dom  |`  i^i  dom  Ord  dom  |`  Ord  i^i  dom
2220, 21ax-mp 7 . . . . 5  Ord 
dom  |`  Ord  i^i  dom
2319, 22syl6ibr 151 . . . 4  dom  Ord  dom  Ord 
dom  |`
24 dmss 4477 . . . . . . . . 9  |`  C_  dom  |`  C_  dom
255, 24ax-mp 7 . . . . . . . 8  dom  |`  C_  dom
26 ssralv 2998 . . . . . . . 8  dom  |` 
C_  dom  dom  dom  `  `  dom  |`  dom  `  `
2725, 26ax-mp 7 . . . . . . 7  dom  dom  `  `  dom  |`  dom  `  `
28 ssralv 2998 . . . . . . . . 9  dom  |` 
C_  dom  dom  `  `  dom  |`  `  `
2925, 28ax-mp 7 . . . . . . . 8  dom  `  `  dom  |`  `  `
3029ralimi 2378 . . . . . . 7  dom  |`  dom  `  `  dom  |`  dom  |`  `  `
3127, 30syl 14 . . . . . 6  dom  dom  `  `  dom  |`  dom  |`  `  `
32 inss1 3151 . . . . . . . . . . . . 13  i^i  dom  C_
3320, 32eqsstri 2969 . . . . . . . . . . . 12  dom  |`  C_
34 simpl 102 . . . . . . . . . . . 12  dom  |`  dom  |`  dom  |`
3533, 34sseldi 2937 . . . . . . . . . . 11  dom  |`  dom  |`
36 fvres 5141 . . . . . . . . . . 11  |`  `
 `
3735, 36syl 14 . . . . . . . . . 10  dom  |`  dom  |`  |`  `
 `
38 simpr 103 . . . . . . . . . . . 12  dom  |`  dom  |`  dom  |`
3933, 38sseldi 2937 . . . . . . . . . . 11  dom  |`  dom  |`
40 fvres 5141 . . . . . . . . . . 11  |`  `
 `
4139, 40syl 14 . . . . . . . . . 10  dom  |`  dom  |`  |`  `
 `
4237, 41eleq12d 2105 . . . . . . . . 9  dom  |`  dom  |`  |`  `
 |`  `
 `
 `
4342imbi2d 219 . . . . . . . 8  dom  |`  dom  |`  |`  `
 |`  `
 `  `
4443ralbidva 2316 . . . . . . 7  dom  |`  dom  |`  |`  `
 |`  `
 dom  |`  `  `
4544ralbiia 2332 . . . . . 6  dom  |`  dom  |`  |`  `
 |`  `
 dom  |`  dom  |`  `  `
4631, 45sylibr 137 . . . . 5  dom  dom  `  `  dom  |`  dom  |`  |`  `
 |`  `
4746a1i 9 . . . 4  dom  dom  dom  `  `  dom  |`  dom  |`  |`  `
 |`  `
4814, 23, 473anim123d 1213 . . 3  dom  : dom  --> On  Ord  dom  dom  dom  `  `  |`  : dom  |`  --> On  Ord  dom  |`  dom  |`  dom  |`  |`  `
 |`  `
49 df-smo 5842 . . 3  Smo  : dom  --> On  Ord  dom  dom  dom  `  `
50 df-smo 5842 . . 3  Smo  |`  |`  : dom  |`  --> On  Ord  dom  |`  dom  |`  dom  |`  |`  `
 |`  `
5148, 49, 503imtr4g 194 . 2  dom  Smo  Smo  |`
5251impcom 116 1  Smo  dom 
Smo  |`
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wral 2300    i^i cin 2910    C_ wss 2911   Ord word 4065   Oncon0 4066   dom cdm 4288   ran crn 4289    |` cres 4290   Fun wfun 4839    Fn wfn 4840   -->wf 4841   ` cfv 4845   Smo wsmo 5841
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842
This theorem is referenced by:  smores3  5849
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