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Theorem smoword 6630
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 6629 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( D  e.  C  <->  ( F `  D )  e.  ( F `  C ) ) )
21notbid 287 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
32ancom2s 779 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
4 smodm2 6619 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
54adantr 453 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
6 simprl 734 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
7 ordelord 4605 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
85, 6, 7syl2anc 644 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
9 simprr 735 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
10 ordelord 4605 . . . 4  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
115, 9, 10syl2anc 644 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
12 ordtri1 4616 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
138, 11, 12syl2anc 644 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
14 simplr 733 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Smo  F )
15 smofvon2 6620 . . . 4  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
16 eloni 4593 . . . 4  |-  ( ( F `  C )  e.  On  ->  Ord  ( F `  C ) )
1714, 15, 163syl 19 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  C ) )
18 smofvon2 6620 . . . 4  |-  ( Smo 
F  ->  ( F `  D )  e.  On )
19 eloni 4593 . . . 4  |-  ( ( F `  D )  e.  On  ->  Ord  ( F `  D ) )
2014, 18, 193syl 19 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  D ) )
21 ordtri1 4616 . . 3  |-  ( ( Ord  ( F `  C )  /\  Ord  ( F `  D ) )  ->  ( ( F `  C )  C_  ( F `  D
)  <->  -.  ( F `  D )  e.  ( F `  C ) ) )
2217, 20, 21syl2anc 644 . 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 278 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 178    /\ wa 360    e. wcel 1726    C_ wss 3322   Ord word 4582   Oncon0 4583    Fn wfn 5451   ` cfv 5456   Smo wsmo 6609
This theorem is referenced by:  cfcoflem  8154  coftr  8155
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-13 1728  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-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  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-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-smo 6610
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