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Theorem smoword 7029
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 7028 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( D  e.  C  <->  ( F `  D )  e.  ( F `  C ) ) )
21notbid 292 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
32ancom2s 800 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( -.  D  e.  C  <->  -.  ( F `  D
)  e.  ( F `
 C ) ) )
4 smodm2 7018 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
54adantr 463 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
6 simprl 754 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
7 ordelord 4889 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
85, 6, 7syl2anc 659 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
9 simprr 755 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
10 ordelord 4889 . . . 4  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
115, 9, 10syl2anc 659 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
12 ordtri1 4900 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
138, 11, 12syl2anc 659 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
14 simplr 753 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Smo  F )
15 smofvon2 7019 . . . 4  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
16 eloni 4877 . . . 4  |-  ( ( F `  C )  e.  On  ->  Ord  ( F `  C ) )
1714, 15, 163syl 20 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  C ) )
18 smofvon2 7019 . . . 4  |-  ( Smo 
F  ->  ( F `  D )  e.  On )
19 eloni 4877 . . . 4  |-  ( ( F `  D )  e.  On  ->  Ord  ( F `  D ) )
2014, 18, 193syl 20 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  D ) )
21 ordtri1 4900 . . 3  |-  ( ( Ord  ( F `  C )  /\  Ord  ( F `  D ) )  ->  ( ( F `  C )  C_  ( F `  D
)  <->  -.  ( F `  D )  e.  ( F `  C ) ) )
2217, 20, 21syl2anc 659 . 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 285 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `  D )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461   Ord word 4866   Oncon0 4867    Fn wfn 5565   ` cfv 5570   Smo wsmo 7008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-smo 7009
This theorem is referenced by:  cfcoflem  8643  coftr  8644
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