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Theorem smoword 6425
Description: A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
Assertion
Ref Expression
smoword  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `  D )
) )

Proof of Theorem smoword
StepHypRef Expression
1 smoord 6424 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( D  e.  C  <->  ( F `  D )  e.  ( F `  C ) ) )
21notbid 285 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
32ancom2s 777 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
4 smodm2 6414 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
54adantr 451 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
6 simprl 732 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
7 ordelord 4451 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
85, 6, 7syl2anc 642 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
9 simprr 733 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
10 ordelord 4451 . . . 4  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
115, 9, 10syl2anc 642 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
12 ordtri1 4462 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
138, 11, 12syl2anc 642 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
14 simplr 731 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Smo  F )
15 smofvon2 6415 . . . 4  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
16 eloni 4439 . . . 4  |-  ( ( F `  C )  e.  On  ->  Ord  ( F `  C ) )
1714, 15, 163syl 18 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  C ) )
18 smofvon2 6415 . . . 4  |-  ( Smo 
F  ->  ( F `  D )  e.  On )
19 eloni 4439 . . . 4  |-  ( ( F `  D )  e.  On  ->  Ord  ( F `  D ) )
2014, 18, 193syl 18 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  D ) )
21 ordtri1 4462 . . 3  |-  ( ( Ord  ( F `  C )  /\  Ord  ( F `  D ) )  ->  ( ( F `  C )  C_  ( F `  D
)  <->  -.  ( F `  D )  e.  ( F `  C ) ) )
2217, 20, 21syl2anc 642 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  C_  ( F `  D )  <->  -.  ( F `  D )  e.  ( F `  C
) ) )
233, 13, 223bitr4d 276 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `  D )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1701    C_ wss 3186   Ord word 4428   Oncon0 4429    Fn wfn 5287   ` cfv 5292   Smo wsmo 6404
This theorem is referenced by:  cfcoflem  7943  coftr  7944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-smo 6405
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