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Theorem smorndom 6385
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 958 . . . . . . 7  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F : A --> B )
2 ffn 5389 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 15 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  F  Fn  A )
4 simpl2 959 . . . . . 6  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Smo  F )
5 smodm2 6372 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
63, 4, 5syl2anc 642 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  A )
7 ordelord 4414 . . . . 5  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
86, 7sylancom 648 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  x )
9 simpl3 960 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  Ord  B )
10 simpr 447 . . . . 5  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  A )
11 smogt 6384 . . . . 5  |-  ( ( F  Fn  A  /\  Smo  F  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
123, 4, 10, 11syl3anc 1182 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  C_  ( F `  x
) )
13 ffvelrn 5663 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
14133ad2antl1 1117 . . . 4  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  ( F `  x )  e.  B )
15 ordtr2 4436 . . . . 5  |-  ( ( Ord  x  /\  Ord  B )  ->  ( (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B )  ->  x  e.  B ) )
1615imp 418 . . . 4  |-  ( ( ( Ord  x  /\  Ord  B )  /\  (
x  C_  ( F `  x )  /\  ( F `  x )  e.  B ) )  ->  x  e.  B )
178, 9, 12, 14, 16syl22anc 1183 . . 3  |-  ( ( ( F : A --> B  /\  Smo  F  /\  Ord  B )  /\  x  e.  A )  ->  x  e.  B )
1817ex 423 . 2  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  ( x  e.  A  ->  x  e.  B ) )
1918ssrdv 3185 1  |-  ( ( F : A --> B  /\  Smo  F  /\  Ord  B
)  ->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   Ord word 4391    Fn wfn 5250   -->wf 5251   ` cfv 5255   Smo wsmo 6362
This theorem is referenced by:  cofsmo  7895  hsmexlem1  8052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-smo 6363
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