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Theorem smorndom 6472
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 5472 . . . . . . 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 6459 . . . . . 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 4496 . . . . 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 6471 . . . . 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 5746 . . . . 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 4518 . . . . 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 3261 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 1710    C_ wss 3228   Ord word 4473    Fn wfn 5332   -->wf 5333   ` cfv 5337   Smo wsmo 6449
This theorem is referenced by:  cofsmo  7985  hsmexlem1  8142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-smo 6450
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