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Theorem smores3 6617
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 5169 . . . . . 6  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 incom 3535 . . . . . 6  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
31, 2eqtri 2458 . . . . 5  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
43eleq2i 2502 . . . 4  |-  ( C  e.  dom  ( A  |`  B )  <->  C  e.  ( dom  A  i^i  B
) )
5 smores 6616 . . . 4  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  dom  ( A  |`  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
64, 5sylan2br 464 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
763adant3 978 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( ( A  |`  B )  |`  C ) )
8 inss2 3564 . . . . . 6  |-  ( dom 
A  i^i  B )  C_  B
98sseli 3346 . . . . 5  |-  ( C  e.  ( dom  A  i^i  B )  ->  C  e.  B )
10 ordelss 4599 . . . . . 6  |-  ( ( Ord  B  /\  C  e.  B )  ->  C  C_  B )
1110ancoms 441 . . . . 5  |-  ( ( C  e.  B  /\  Ord  B )  ->  C  C_  B )
129, 11sylan 459 . . . 4  |-  ( ( C  e.  ( dom 
A  i^i  B )  /\  Ord  B )  ->  C  C_  B )
13123adant1 976 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  C  C_  B
)
14 resabs1 5177 . . 3  |-  ( C 
C_  B  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  C )
)
15 smoeq 6614 . . 3  |-  ( ( ( A  |`  B )  |`  C )  =  ( A  |`  C )  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
1613, 14, 153syl 19 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
177, 16mpbid 203 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   Ord word 4582   dom cdm 4880    |` cres 4882   Smo wsmo 6609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-smo 6610
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