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Theorem cfsmo 7897
Description: The map in cff1 7884 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cfsmo  |-  ( A  e.  On  ->  E. f
( f : ( cf `  A ) --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
C_  ( f `  w ) ) )
Distinct variable group:    A, f, w, z

Proof of Theorem cfsmo
Dummy variables  m  h  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 4879 . . . . 5  |-  ( x  =  z  ->  dom  x  =  dom  z )
21fveq2d 5529 . . . 4  |-  ( x  =  z  ->  (
h `  dom  x )  =  ( h `  dom  z ) )
3 fveq2 5525 . . . . . . 7  |-  ( n  =  m  ->  (
x `  n )  =  ( x `  m ) )
4 suceq 4457 . . . . . . 7  |-  ( ( x `  n )  =  ( x `  m )  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
53, 4syl 15 . . . . . 6  |-  ( n  =  m  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
65cbviunv 3941 . . . . 5  |-  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  x  suc  ( x `  m )
7 fveq1 5524 . . . . . . 7  |-  ( x  =  z  ->  (
x `  m )  =  ( z `  m ) )
8 suceq 4457 . . . . . . 7  |-  ( ( x `  m )  =  ( z `  m )  ->  suc  ( x `  m
)  =  suc  (
z `  m )
)
97, 8syl 15 . . . . . 6  |-  ( x  =  z  ->  suc  ( x `  m
)  =  suc  (
z `  m )
)
101, 9iuneq12d 3929 . . . . 5  |-  ( x  =  z  ->  U_ m  e.  dom  x  suc  (
x `  m )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
116, 10syl5eq 2327 . . . 4  |-  ( x  =  z  ->  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
122, 11uneq12d 3330 . . 3  |-  ( x  =  z  ->  (
( h `  dom  x )  u.  U_ n  e.  dom  x  suc  ( x `  n
) )  =  ( ( h `  dom  z )  u.  U_ m  e.  dom  z  suc  ( z `  m
) ) )
1312cbvmptv 4111 . 2  |-  ( x  e.  _V  |->  ( ( h `  dom  x
)  u.  U_ n  e.  dom  x  suc  (
x `  n )
) )  =  ( z  e.  _V  |->  ( ( h `  dom  z )  u.  U_ m  e.  dom  z  suc  ( z `  m
) ) )
14 eqid 2283 . 2  |-  (recs ( ( x  e.  _V  |->  ( ( h `  dom  x )  u.  U_ n  e.  dom  x  suc  ( x `  n
) ) ) )  |`  ( cf `  A
) )  =  (recs ( ( x  e. 
_V  |->  ( ( h `
 dom  x )  u.  U_ n  e.  dom  x  suc  ( x `  n ) ) ) )  |`  ( cf `  A ) )
1513, 14cfsmolem 7896 1  |-  ( A  e.  On  ->  E. f
( f : ( cf `  A ) --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
C_  ( f `  w ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    C_ wss 3152   U_ciun 3905    e. cmpt 4077   Oncon0 4392   suc csuc 4394   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255   Smo wsmo 6362  recscrecs 6387   cfccf 7570
This theorem is referenced by:  cfidm  7901  pwcfsdom  8205
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-rep 4131  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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572  df-cf 7574  df-acn 7575
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