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Theorem smoord 6627
Description: A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
Assertion
Ref Expression
smoord  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )

Proof of Theorem smoord
StepHypRef Expression
1 smodm2 6617 . . . 4  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
21adantr 452 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
3 simprl 733 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
4 ordelord 4603 . . 3  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
52, 3, 4syl2anc 643 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
6 simprr 734 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
7 ordelord 4603 . . 3  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
82, 6, 7syl2anc 643 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
9 ordtri3or 4613 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  e.  D  \/  C  =  D  \/  D  e.  C ) )
10 simp3 959 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  C  e.  D )
11 smoel2 6625 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  D
) )  ->  ( F `  C )  e.  ( F `  D
) )
1211expr 599 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  D  e.  A )  ->  ( C  e.  D  ->  ( F `  C
)  e.  ( F `
 D ) ) )
1312adantrl 697 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  ->  ( F `  C )  e.  ( F `  D ) ) )
14133impia 1150 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  ( F `  C )  e.  ( F `  D
) )
1510, 142thd 232 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
16153expia 1155 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
17 ordirr 4599 . . . . . . . . 9  |-  ( Ord 
C  ->  -.  C  e.  C )
185, 17syl 16 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  -.  C  e.  C )
19183adant3 977 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  C  e.  C )
20 simp3 959 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  C  =  D )
2119, 20neleqtrd 2531 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  C  e.  D )
22 smofvon2 6618 . . . . . . . . . 10  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
2322ad2antlr 708 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  On )
24 eloni 4591 . . . . . . . . 9  |-  ( ( F `  C )  e.  On  ->  Ord  ( F `  C ) )
25 ordirr 4599 . . . . . . . . 9  |-  ( Ord  ( F `  C
)  ->  -.  ( F `  C )  e.  ( F `  C
) )
2623, 24, 253syl 19 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  -.  ( F `  C )  e.  ( F `  C ) )
27263adant3 977 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  ( F `  C )  e.  ( F `  C ) )
2820fveq2d 5732 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  ( F `  C )  =  ( F `  D ) )
2927, 28neleqtrd 2531 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  ( F `  C )  e.  ( F `  D ) )
3021, 292falsed 341 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
31303expia 1155 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  =  D  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
3283adant3 977 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  Ord  D )
33 ordn2lp 4601 . . . . . . . 8  |-  ( Ord 
D  ->  -.  ( D  e.  C  /\  C  e.  D )
)
3432, 33syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( D  e.  C  /\  C  e.  D
) )
35 pm3.2 435 . . . . . . . 8  |-  ( D  e.  C  ->  ( C  e.  D  ->  ( D  e.  C  /\  C  e.  D )
) )
36353ad2ant3 980 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( C  e.  D  ->  ( D  e.  C  /\  C  e.  D )
) )
3734, 36mtod 170 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  C  e.  D )
3823, 24syl 16 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  C ) )
39383adant3 977 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  Ord  ( F `  C ) )
40 ordn2lp 4601 . . . . . . . 8  |-  ( Ord  ( F `  C
)  ->  -.  (
( F `  C
)  e.  ( F `
 D )  /\  ( F `  D )  e.  ( F `  C ) ) )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) )
42 smoel2 6625 . . . . . . . . . 10  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  C
) )  ->  ( F `  D )  e.  ( F `  C
) )
4342adantrlr 704 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( ( C  e.  A  /\  D  e.  A )  /\  D  e.  C ) )  -> 
( F `  D
)  e.  ( F `
 C ) )
44433impb 1149 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( F `  D )  e.  ( F `  C
) )
45 pm3.21 436 . . . . . . . 8  |-  ( ( F `  D )  e.  ( F `  C )  ->  (
( F `  C
)  e.  ( F `
 D )  -> 
( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) ) )
4644, 45syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  (
( F `  C
)  e.  ( F `
 D )  -> 
( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) ) )
4741, 46mtod 170 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( F `  C )  e.  ( F `  D ) )
4837, 472falsed 341 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
49483expia 1155 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( D  e.  C  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
5016, 31, 493jaod 1248 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( C  e.  D  \/  C  =  D  \/  D  e.  C
)  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
519, 50syl5 30 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( Ord  C  /\  Ord  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
525, 8, 51mp2and 661 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1652    e. wcel 1725   Ord word 4580   Oncon0 4581    Fn wfn 5449   ` cfv 5454   Smo wsmo 6607
This theorem is referenced by:  smoword  6628  smoiso2  6631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-smo 6608
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