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Theorem smorndom 6633
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 961 . . . . . . 7  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F : A --> B )
2 ffn 5594 . . . . . . 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 962 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Smo  F )
5 smodm2 6620 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
63, 4, 5syl2anc 644 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  A )
7 ordelord 4606 . . . . 5  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
86, 7sylancom 650 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  x )
9 simpl3 963 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  B )
10 simpr 449 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  A )
11 smogt 6632 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
123, 4, 10, 11syl3anc 1185 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
13 ffvelrn 5871 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
14133ad2antl1 1120 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  ( F `  x )  e.  B )
15 ordtr2 4628 . . . . 5  |-  ( ( Ord  x  /\  Ord  B )  ->  ( (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B )  ->  x  e.  B ) )
1615imp 420 . . . 4  |-  ( ( ( Ord  x  /\  Ord  B )  /\  (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B ) )  ->  x  e.  B )
178, 9, 12, 14, 16syl22anc 1186 . . 3  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  B )
1817ex 425 . 2  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  ( x  e.  A  ->  x  e.  B ) )
1918ssrdv 3356 1  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726    C_ wss 3322   Ord word 4583    Fn wfn 5452   -->wf 5453   ` cfv 5457   Smo wsmo 6610
This theorem is referenced by:  cofsmo  8154  hsmexlem1  8311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-smo 6611
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