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Theorem smores3 6058
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
smores3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( A  |`  C ) )

Proof of Theorem smores3
StepHypRef Expression
1 dmres 4734 . . . . . 6  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 incom 3192 . . . . . 6  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
31, 2eqtri 2108 . . . . 5  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
43eleq2i 2154 . . . 4  |-  ( C  e.  dom  ( A  |`  B )  <->  C  e.  ( dom  A  i^i  B
) )
5 smores 6057 . . . 4  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  dom  ( A  |`  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
64, 5sylan2br 282 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
763adant3 963 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( ( A  |`  B )  |`  C ) )
8 inss2 3221 . . . . . 6  |-  ( dom 
A  i^i  B )  C_  B
98sseli 3021 . . . . 5  |-  ( C  e.  ( dom  A  i^i  B )  ->  C  e.  B )
10 ordelss 4206 . . . . . 6  |-  ( ( Ord  B  /\  C  e.  B )  ->  C  C_  B )
1110ancoms 264 . . . . 5  |-  ( ( C  e.  B  /\  Ord  B )  ->  C  C_  B )
129, 11sylan 277 . . . 4  |-  ( ( C  e.  ( dom 
A  i^i  B )  /\  Ord  B )  ->  C  C_  B )
13123adant1 961 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  C  C_  B
)
14 resabs1 4742 . . 3  |-  ( C 
C_  B  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  C )
)
15 smoeq 6055 . . 3  |-  ( ( ( A  |`  B )  |`  C )  =  ( A  |`  C )  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
1613, 14, 153syl 17 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
177, 16mpbid 145 1  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438    i^i cin 2998    C_ wss 2999   Ord word 4189   dom cdm 4438    |` cres 4440   Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-tr 3937  df-iord 4193  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-smo 6051
This theorem is referenced by: (None)
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