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Theorem smorndom 6589
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 960 . . . . . . 7  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F : A --> B )
2 ffn 5550 . . . . . . 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 961 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Smo  F )
5 smodm2 6576 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
63, 4, 5syl2anc 643 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  A )
7 ordelord 4563 . . . . 5  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
86, 7sylancom 649 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  x )
9 simpl3 962 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  B )
10 simpr 448 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  A )
11 smogt 6588 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
123, 4, 10, 11syl3anc 1184 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
13 ffvelrn 5827 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
14133ad2antl1 1119 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  ( F `  x )  e.  B )
15 ordtr2 4585 . . . . 5  |-  ( ( Ord  x  /\  Ord  B )  ->  ( (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B )  ->  x  e.  B ) )
1615imp 419 . . . 4  |-  ( ( ( Ord  x  /\  Ord  B )  /\  (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B ) )  ->  x  e.  B )
178, 9, 12, 14, 16syl22anc 1185 . . 3  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  B )
1817ex 424 . 2  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  ( x  e.  A  ->  x  e.  B ) )
1918ssrdv 3314 1  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721    C_ wss 3280   Ord word 4540    Fn wfn 5408   -->wf 5409   ` cfv 5413   Smo wsmo 6566
This theorem is referenced by:  cofsmo  8105  hsmexlem1  8262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-smo 6567
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