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Theorem smores2 7078
Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Assertion
Ref Expression
smores2  |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )

Proof of Theorem smores2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 7071 . . . . . . 7  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
21simp1bi 1024 . . . . . 6  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffun 5736 . . . . . 6  |-  ( F : dom  F --> On  ->  Fun 
F )
42, 3syl 17 . . . . 5  |-  ( Smo 
F  ->  Fun  F )
5 funres 5624 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
6 funfn 5614 . . . . . 6  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
75, 6sylib 200 . . . . 5  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
84, 7syl 17 . . . 4  |-  ( Smo 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
9 df-ima 4850 . . . . . 6  |-  ( F
" A )  =  ran  ( F  |`  A )
10 imassrn 5182 . . . . . 6  |-  ( F
" A )  C_  ran  F
119, 10eqsstr3i 3465 . . . . 5  |-  ran  ( F  |`  A )  C_  ran  F
12 frn 5740 . . . . . 6  |-  ( F : dom  F --> On  ->  ran 
F  C_  On )
132, 12syl 17 . . . . 5  |-  ( Smo 
F  ->  ran  F  C_  On )
1411, 13syl5ss 3445 . . . 4  |-  ( Smo 
F  ->  ran  ( F  |`  A )  C_  On )
15 df-f 5589 . . . 4  |-  ( ( F  |`  A ) : dom  ( F  |`  A ) --> On  <->  ( ( F  |`  A )  Fn 
dom  ( F  |`  A )  /\  ran  ( F  |`  A ) 
C_  On ) )
168, 14, 15sylanbrc 671 . . 3  |-  ( Smo 
F  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> On )
1716adantr 467 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  ( F  |`  A ) : dom  ( F  |`  A ) --> On )
18 smodm 7075 . . 3  |-  ( Smo 
F  ->  Ord  dom  F
)
19 ordin 5456 . . . . 5  |-  ( ( Ord  A  /\  Ord  dom 
F )  ->  Ord  ( A  i^i  dom  F
) )
20 dmres 5128 . . . . . 6  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21 ordeq 5433 . . . . . 6  |-  ( dom  ( F  |`  A )  =  ( A  i^i  dom 
F )  ->  ( Ord  dom  ( F  |`  A )  <->  Ord  ( A  i^i  dom  F )
) )
2220, 21ax-mp 5 . . . . 5  |-  ( Ord 
dom  ( F  |`  A )  <->  Ord  ( A  i^i  dom  F )
)
2319, 22sylibr 216 . . . 4  |-  ( ( Ord  A  /\  Ord  dom 
F )  ->  Ord  dom  ( F  |`  A ) )
2423ancoms 455 . . 3  |-  ( ( Ord  dom  F  /\  Ord  A )  ->  Ord  dom  ( F  |`  A ) )
2518, 24sylan 474 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  Ord  dom  ( F  |`  A ) )
26 resss 5131 . . . . . 6  |-  ( F  |`  A )  C_  F
27 dmss 5037 . . . . . 6  |-  ( ( F  |`  A )  C_  F  ->  dom  ( F  |`  A )  C_  dom  F )
2826, 27ax-mp 5 . . . . 5  |-  dom  ( F  |`  A )  C_  dom  F
291simp3bi 1026 . . . . 5  |-  ( Smo 
F  ->  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
30 ssralv 3495 . . . . 5  |-  ( dom  ( F  |`  A ) 
C_  dom  F  ->  ( A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
3128, 29, 30mpsyl 65 . . . 4  |-  ( Smo 
F  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
3231adantr 467 . . 3  |-  ( ( Smo  F  /\  Ord  A )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) )
33 ordtr1 5469 . . . . . . . . . . 11  |-  ( Ord 
dom  ( F  |`  A )  ->  (
( y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  ->  y  e.  dom  ( F  |`  A ) ) )
3425, 33syl 17 . . . . . . . . . 10  |-  ( ( Smo  F  /\  Ord  A )  ->  ( (
y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  -> 
y  e.  dom  ( F  |`  A ) ) )
35 inss1 3654 . . . . . . . . . . . 12  |-  ( A  i^i  dom  F )  C_  A
3620, 35eqsstri 3464 . . . . . . . . . . 11  |-  dom  ( F  |`  A )  C_  A
3736sseli 3430 . . . . . . . . . 10  |-  ( y  e.  dom  ( F  |`  A )  ->  y  e.  A )
3834, 37syl6 34 . . . . . . . . 9  |-  ( ( Smo  F  /\  Ord  A )  ->  ( (
y  e.  x  /\  x  e.  dom  ( F  |`  A ) )  -> 
y  e.  A ) )
3938expcomd 440 . . . . . . . 8  |-  ( ( Smo  F  /\  Ord  A )  ->  ( x  e.  dom  ( F  |`  A )  ->  (
y  e.  x  -> 
y  e.  A ) ) )
4039imp31 434 . . . . . . 7  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  y  e.  A )
41 fvres 5884 . . . . . . 7  |-  ( y  e.  A  ->  (
( F  |`  A ) `
 y )  =  ( F `  y
) )
4240, 41syl 17 . . . . . 6  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( F  |`  A ) `  y
)  =  ( F `
 y ) )
4336sseli 3430 . . . . . . . 8  |-  ( x  e.  dom  ( F  |`  A )  ->  x  e.  A )
44 fvres 5884 . . . . . . . 8  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
4543, 44syl 17 . . . . . . 7  |-  ( x  e.  dom  ( F  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
4645ad2antlr 734 . . . . . 6  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
4742, 46eleq12d 2525 . . . . 5  |-  ( ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  /\  y  e.  x )  ->  ( ( ( F  |`  A ) `  y
)  e.  ( ( F  |`  A ) `  x )  <->  ( F `  y )  e.  ( F `  x ) ) )
4847ralbidva 2826 . . . 4  |-  ( ( ( Smo  F  /\  Ord  A )  /\  x  e.  dom  ( F  |`  A ) )  -> 
( A. y  e.  x  ( ( F  |`  A ) `  y
)  e.  ( ( F  |`  A ) `  x )  <->  A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
4948ralbidva 2826 . . 3  |-  ( ( Smo  F  /\  Ord  A )  ->  ( A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x )  <->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
5032, 49mpbird 236 . 2  |-  ( ( Smo  F  /\  Ord  A )  ->  A. x  e.  dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x ) )
51 dfsmo2 7071 . 2  |-  ( Smo  ( F  |`  A )  <-> 
( ( F  |`  A ) : dom  ( F  |`  A ) --> On  /\  Ord  dom  ( F  |`  A )  /\  A. x  e. 
dom  ( F  |`  A ) A. y  e.  x  ( ( F  |`  A ) `  y )  e.  ( ( F  |`  A ) `
 x ) ) )
5217, 25, 50, 51syl3anbrc 1193 1  |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739    i^i cin 3405    C_ wss 3406   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Ord word 5425   Oncon0 5426   Fun wfun 5579    Fn wfn 5580   -->wf 5581   ` cfv 5585   Smo wsmo 7069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-tr 4501  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-smo 7070
This theorem is referenced by: (None)
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