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Theorem cfsmo 8107
Description: The map in cff1 8094 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 5029 . . . . 5  |-  ( x  =  z  ->  dom  x  =  dom  z )
21fveq2d 5691 . . . 4  |-  ( x  =  z  ->  (
h `  dom  x )  =  ( h `  dom  z ) )
3 fveq2 5687 . . . . . . 7  |-  ( n  =  m  ->  (
x `  n )  =  ( x `  m ) )
4 suceq 4606 . . . . . . 7  |-  ( ( x `  n )  =  ( x `  m )  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
53, 4syl 16 . . . . . 6  |-  ( n  =  m  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
65cbviunv 4090 . . . . 5  |-  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  x  suc  ( x `  m )
7 fveq1 5686 . . . . . . 7  |-  ( x  =  z  ->  (
x `  m )  =  ( z `  m ) )
8 suceq 4606 . . . . . . 7  |-  ( ( x `  m )  =  ( z `  m )  ->  suc  ( x `  m
)  =  suc  (
z `  m )
)
97, 8syl 16 . . . . . 6  |-  ( x  =  z  ->  suc  ( x `  m
)  =  suc  (
z `  m )
)
101, 9iuneq12d 4077 . . . . 5  |-  ( x  =  z  ->  U_ m  e.  dom  x  suc  (
x `  m )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
116, 10syl5eq 2448 . . . 4  |-  ( x  =  z  ->  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
122, 11uneq12d 3462 . . 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 4260 . 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 2404 . 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 8106 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 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    u. cun 3278    C_ wss 3280   U_ciun 4053    e. cmpt 4226   Oncon0 4541   suc csuc 4543   dom cdm 4837    |` cres 4839   -->wf 5409   ` cfv 5413   Smo wsmo 6566  recscrecs 6591   cfccf 7780
This theorem is referenced by:  cfidm  8111  pwcfsdom  8414
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-rep 4280  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-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  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-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-smo 6567  df-recs 6592  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-card 7782  df-cf 7784  df-acn 7785
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