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Theorem smores2 7015
Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Assertion
Ref Expression
smores2  |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )

Proof of Theorem smores2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 7008 . . . . . . 7  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
21simp1bi 1006 . . . . . 6  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffun 5724 . . . . . 6  |-  ( F : dom  F --> On  ->  Fun 
F )
42, 3syl 16 . . . . 5  |-  ( Smo 
F  ->  Fun  F )
5 funres 5618 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funfn 5608 . . . . . 6  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
75, 6sylib 196 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
84, 7syl 16 . . . 4  |-  ( Smo 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
9 df-ima 5005 . . . . . 6  |-  ( F
" A )  =  ran  ( F  |`  A )
10 imassrn 5339 . . . . . 6  |-  ( F
" A )  C_  ran  F
119, 10eqsstr3i 3528 . . . . 5  |-  ran  ( F  |`  A )  C_  ran  F
12 frn 5728 . . . . . 6  |-  ( F : dom  F --> On  ->  ran 
F  C_  On )
132, 12syl 16 . . . . 5  |-  ( Smo 
F  ->  ran  F  C_  On )
1411, 13syl5ss 3508 . . . 4  |-  ( Smo 
F  ->  ran  ( F  |`  A )  C_  On )
15 df-f 5583 . . . 4  |-  ( ( F  |`  A ) : dom  ( F  |`  A ) --> On  <->  ( ( F  |`  A )  Fn 
dom  ( F  |`  A )  /\  ran  ( F  |`  A ) 
C_  On ) )
168, 14, 15sylanbrc 664 . . 3  |-  ( Smo 
F  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> On )
1716adantr 465 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> On )
18 smodm 7012 . . 3  |-  ( Smo 
F  ->  Ord  dom  F
)
19 ordin 4901 . . . . 5  |-  ( ( Ord  A  /\  Ord  dom 
F )  ->  Ord  ( A  i^i  dom  F
) )
20 dmres 5285 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21 ordeq 4878 . . . . . 6  |-  ( dom  ( F  |`  A )  =  ( A  i^i  dom 
F )  ->  ( Ord  dom  ( F  |`  A )  <->  Ord  ( A  i^i  dom  F )
) )
2220, 21ax-mp 5 . . . . 5  |-  ( Ord 
dom  ( F  |`  A )  <->  Ord  ( A  i^i  dom  F )
)
2319, 22sylibr 212 . . . 4  |-  ( ( Ord  A  /\  Ord  dom 
F )  ->  Ord  dom  ( F  |`  A ) )
2423ancoms 453 . . 3  |-  ( ( Ord  dom  F  /\  Ord  A )  ->  Ord  dom  ( F  |`  A ) )
2518, 24sylan 471 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  Ord  dom  ( F  |`  A ) )
26 resss 5288 . . . . . 6  |-  ( F  |`  A )  C_  F
27 dmss 5193 . . . . . 6  |-  ( ( F  |`  A )  C_  F  ->  dom  ( F  |`  A )  C_  dom  F )
2826, 27ax-mp 5 . . . . 5  |-  dom  ( F  |`  A )  C_  dom  F
291simp3bi 1008 . . . . 5  |-  ( Smo 
F  ->  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
30 ssralv 3557 . . . . 5  |-  ( dom  ( F  |`  A ) 
C_  dom  F  ->  ( A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
3128, 29, 30mpsyl 63 . . . 4  |-  ( Smo 
F  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
3231adantr 465 . . 3  |-  ( ( Smo  F  /\  Ord  A )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
33 ordtr1 4914 . . . . . . . . . . 11  |-  ( Ord 
dom  ( F  |`  A )  ->  (
( y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  ->  y  e.  dom  ( F  |`  A ) ) )
3425, 33syl 16 . . . . . . . . . 10  |-  ( ( Smo  F  /\  Ord  A )  ->  ( (
y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  -> 
y  e.  dom  ( F  |`  A ) ) )
35 inss1 3711 . . . . . . . . . . . 12  |-  ( A  i^i  dom  F )  C_  A
3620, 35eqsstri 3527 . . . . . . . . . . 11  |-  dom  ( F  |`  A )  C_  A
3736sseli 3493 . . . . . . . . . 10  |-  ( y  e.  dom  ( F  |`  A )  ->  y  e.  A )
3834, 37syl6 33 . . . . . . . . 9  |-  ( ( Smo  F  /\  Ord  A )  ->  ( (
y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  -> 
y  e.  A ) )
3938expcomd 438 . . . . . . . 8  |-  ( ( Smo  F  /\  Ord  A )  ->  ( x  e.  dom  ( F  |`  A )  ->  (
y  e.  x  -> 
y  e.  A ) ) )
4039imp31 432 . . . . . . 7  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  y  e.  A )
41 fvres 5871 . . . . . . 7  |-  ( y  e.  A  ->  (
( F  |`  A ) `
 y )  =  ( F `  y
) )
4240, 41syl 16 . . . . . 6  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( F  |`  A ) `  y
)  =  ( F `
 y ) )
4336sseli 3493 . . . . . . . 8  |-  ( x  e.  dom  ( F  |`  A )  ->  x  e.  A )
44 fvres 5871 . . . . . . . 8  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
4543, 44syl 16 . . . . . . 7  |-  ( x  e.  dom  ( F  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
4645ad2antlr 726 . . . . . 6  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
4742, 46eleq12d 2542 . . . . 5  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( ( F  |`  A ) `  y
)  e.  ( ( F  |`  A ) `  x )  <->  ( F `  y )  e.  ( F `  x ) ) )
4847ralbidva 2893 . . . 4  |-  ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  -> 
( A. y  e.  x  ( ( F  |`  A ) `  y
)  e.  ( ( F  |`  A ) `  x )  <->  A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
4948ralbidva 2893 . . 3  |-  ( ( Smo  F  /\  Ord  A )  ->  ( A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x )  <->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
5032, 49mpbird 232 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x ) )
51 dfsmo2 7008 . 2  |-  ( Smo  ( F  |`  A )  <-> 
( ( F  |`  A ) : dom  ( F  |`  A ) --> On  /\  Ord  dom  ( F  |`  A )  /\  A. x  e. 
dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x ) ) )
5217, 25, 50, 51syl3anbrc 1175 1  |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807    i^i cin 3468    C_ wss 3469   Ord word 4870   Oncon0 4871   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573    Fn wfn 5574   -->wf 5575   ` cfv 5579   Smo wsmo 7006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-smo 7007
This theorem is referenced by: (None)
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