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Theorem smores3 6237
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 4886 . . . . . 6  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 incom 3299 . . . . . 6  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
31, 2eqtri 2178 . . . . 5  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
43eleq2i 2224 . . . 4  |-  ( C  e.  dom  ( A  |`  B )  <->  C  e.  ( dom  A  i^i  B
) )
5 smores 6236 . . . 4  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  dom  ( A  |`  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
64, 5sylan2br 286 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
763adant3 1002 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( ( A  |`  B )  |`  C ) )
8 inss2 3328 . . . . . 6  |-  ( dom 
A  i^i  B )  C_  B
98sseli 3124 . . . . 5  |-  ( C  e.  ( dom  A  i^i  B )  ->  C  e.  B )
10 ordelss 4339 . . . . . 6  |-  ( ( Ord  B  /\  C  e.  B )  ->  C  C_  B )
1110ancoms 266 . . . . 5  |-  ( ( C  e.  B  /\  Ord  B )  ->  C  C_  B )
129, 11sylan 281 . . . 4  |-  ( ( C  e.  ( dom 
A  i^i  B )  /\  Ord  B )  ->  C  C_  B )
13123adant1 1000 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  C  C_  B
)
14 resabs1 4894 . . 3  |-  ( C 
C_  B  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  C )
)
15 smoeq 6234 . . 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 146 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 104    /\ w3a 963    = wceq 1335    e. wcel 2128    i^i cin 3101    C_ wss 3102   Ord word 4322   dom cdm 4585    |` cres 4587   Smo wsmo 6229
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-iord 4326  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-smo 6230
This theorem is referenced by: (None)
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