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Theorem smoword 6383
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 6382 . . . 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 6372 . . . . 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 4414 . . . 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 4414 . . . 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 4425 . . 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 6373 . . . 4  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
16 eloni 4402 . . . 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 6373 . . . 4  |-  ( Smo 
F  ->  ( F `  D )  e.  On )
19 eloni 4402 . . . 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 4425 . . 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 1684    C_ wss 3152   Ord word 4391   Oncon0 4392    Fn wfn 5250   ` cfv 5255   Smo wsmo 6362
This theorem is referenced by:  cfcoflem  7898  coftr  7899
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-13 1686  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-pow 4188  ax-pr 4214
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 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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  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-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-smo 6363
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