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Theorem smoword 6932
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 6931 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  A
) )  ->  ( D  e.  C  <->  ( F `  D )  e.  ( F `  C ) ) )
21notbid 294 . . 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 6921 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
54adantr 465 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
6 simprl 755 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
7 ordelord 4844 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
85, 6, 7syl2anc 661 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
9 simprr 756 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
10 ordelord 4844 . . . 4  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
115, 9, 10syl2anc 661 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
12 ordtri1 4855 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
138, 11, 12syl2anc 661 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  C_  D  <->  -.  D  e.  C ) )
14 simplr 754 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Smo  F )
15 smofvon2 6922 . . . 4  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
16 eloni 4832 . . . 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 6922 . . . 4  |-  ( Smo 
F  ->  ( F `  D )  e.  On )
19 eloni 4832 . . . 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 4855 . . 3  |-  ( ( Ord  ( F `  C )  /\  Ord  ( F `  D ) )  ->  ( ( F `  C )  C_  ( F `  D
)  <->  -.  ( F `  D )  e.  ( F `  C ) ) )
2217, 20, 21syl2anc 661 . 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 369    e. wcel 1758    C_ wss 3431   Ord word 4821   Oncon0 4822    Fn wfn 5516   ` cfv 5521   Smo wsmo 6911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fv 5529  df-smo 6912
This theorem is referenced by:  cfcoflem  8547  coftr  8548
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