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Theorem cfsmo 8152
Description: The map in cff1 8139 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 5071 . . . . 5  |-  ( x  =  z  ->  dom  x  =  dom  z )
21fveq2d 5733 . . . 4  |-  ( x  =  z  ->  (
h `  dom  x )  =  ( h `  dom  z ) )
3 fveq2 5729 . . . . . . 7  |-  ( n  =  m  ->  (
x `  n )  =  ( x `  m ) )
4 suceq 4647 . . . . . . 7  |-  ( ( x `  n )  =  ( x `  m )  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
53, 4syl 16 . . . . . 6  |-  ( n  =  m  ->  suc  ( x `  n
)  =  suc  (
x `  m )
)
65cbviunv 4131 . . . . 5  |-  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  x  suc  ( x `  m )
7 fveq1 5728 . . . . . . 7  |-  ( x  =  z  ->  (
x `  m )  =  ( z `  m ) )
8 suceq 4647 . . . . . . 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 4118 . . . . 5  |-  ( x  =  z  ->  U_ m  e.  dom  x  suc  (
x `  m )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
116, 10syl5eq 2481 . . . 4  |-  ( x  =  z  ->  U_ n  e.  dom  x  suc  (
x `  n )  =  U_ m  e.  dom  z  suc  ( z `  m ) )
122, 11uneq12d 3503 . . 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 4301 . 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 2437 . 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 8151 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 937   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   _Vcvv 2957    u. cun 3319    C_ wss 3321   U_ciun 4094    e. cmpt 4267   Oncon0 4582   suc csuc 4584   dom cdm 4879    |` cres 4881   -->wf 5451   ` cfv 5455   Smo wsmo 6608  recscrecs 6633   cfccf 7825
This theorem is referenced by:  cfidm  8156  pwcfsdom  8459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-smo 6609  df-recs 6634  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-card 7827  df-cf 7829  df-acn 7830
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