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Theorem smores3 6370
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 4976 . . . . . 6  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 incom 3361 . . . . . 6  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
31, 2eqtri 2303 . . . . 5  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
43eleq2i 2347 . . . 4  |-  ( C  e.  dom  ( A  |`  B )  <->  C  e.  ( dom  A  i^i  B
) )
5 smores 6369 . . . 4  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  dom  ( A  |`  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
64, 5sylan2br 462 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B ) )  ->  Smo  ( ( A  |`  B )  |`  C ) )
763adant3 975 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  Smo  ( ( A  |`  B )  |`  C ) )
8 inss2 3390 . . . . . 6  |-  ( dom 
A  i^i  B )  C_  B
98sseli 3176 . . . . 5  |-  ( C  e.  ( dom  A  i^i  B )  ->  C  e.  B )
10 ordelss 4408 . . . . . 6  |-  ( ( Ord  B  /\  C  e.  B )  ->  C  C_  B )
1110ancoms 439 . . . . 5  |-  ( ( C  e.  B  /\  Ord  B )  ->  C  C_  B )
129, 11sylan 457 . . . 4  |-  ( ( C  e.  ( dom 
A  i^i  B )  /\  Ord  B )  ->  C  C_  B )
13123adant1 973 . . 3  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  C  C_  B
)
14 resabs1 4984 . . 3  |-  ( C 
C_  B  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  C )
)
15 smoeq 6367 . . 3  |-  ( ( ( A  |`  B )  |`  C )  =  ( A  |`  C )  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
1613, 14, 153syl 18 . 2  |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord  B
)  ->  ( Smo  ( ( A  |`  B )  |`  C )  <->  Smo  ( A  |`  C ) ) )
177, 16mpbid 201 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   Ord word 4391   dom cdm 4689    |` cres 4691   Smo wsmo 6362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-smo 6363
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