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Theorem smorndom 6829
Description: The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
Assertion
Ref Expression
smorndom  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  A  C_  B
)

Proof of Theorem smorndom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 991 . . . . . . 7  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F : A --> B )
2 ffn 5559 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F  Fn  A )
4 simpl2 992 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Smo  F )
5 smodm2 6816 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
63, 4, 5syl2anc 661 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  A )
7 ordelord 4741 . . . . 5  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
86, 7sylancom 667 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  x )
9 simpl3 993 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  B )
10 simpr 461 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  A )
11 smogt 6828 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
123, 4, 10, 11syl3anc 1218 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
13 ffvelrn 5841 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
14133ad2antl1 1150 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  ( F `  x )  e.  B )
15 ordtr2 4763 . . . . 5  |-  ( ( Ord  x  /\  Ord  B )  ->  ( (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B )  ->  x  e.  B ) )
1615imp 429 . . . 4  |-  ( ( ( Ord  x  /\  Ord  B )  /\  (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B ) )  ->  x  e.  B )
178, 9, 12, 14, 16syl22anc 1219 . . 3  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  B )
1817ex 434 . 2  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  ( x  e.  A  ->  x  e.  B ) )
1918ssrdv 3362 1  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  A  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756    C_ wss 3328   Ord word 4718    Fn wfn 5413   -->wf 5414   ` cfv 5418   Smo wsmo 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-smo 6807
This theorem is referenced by:  cofsmo  8438  hsmexlem1  8595
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