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Theorem tfrlem10 6403
Description: Lemma for transfinite recursion. We define class  C by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to,  On. Using this assumption we will prove facts about  C that will lead to a contradiction in tfrlem14 6407, thus showing the domain of recs does in fact equal  On. Here we show (under the false assumption) that  C is a function extending the domain of recs
( F ) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlem.3  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
Assertion
Ref Expression
tfrlem10  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
Distinct variable groups:    x, f,
y, C    f, F, x, y
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem10
StepHypRef Expression
1 fvex 5539 . . . . . . 7  |-  ( F `
recs ( F ) )  e.  _V
2 funsng 5298 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  ( F ` recs ( F
) )  e.  _V )  ->  Fun  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
31, 2mpan2 652 . . . . . 6  |-  ( dom recs
( F )  e.  On  ->  Fun  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
4 tfrlem.1 . . . . . . 7  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
54tfrlem7 6399 . . . . . 6  |-  Fun recs ( F )
63, 5jctil 523 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( Fun recs ( F )  /\  Fun  {
<. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
71dmsnop 5147 . . . . . . 7  |-  dom  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. }  =  { dom recs ( F ) }
87ineq2i 3367 . . . . . 6  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  i^i  { dom recs ( F ) } )
94tfrlem8 6400 . . . . . . 7  |-  Ord  dom recs ( F )
10 orddisj 4430 . . . . . . 7  |-  ( Ord 
dom recs ( F )  -> 
( dom recs ( F
)  i^i  { dom recs ( F ) } )  =  (/) )
119, 10ax-mp 8 . . . . . 6  |-  ( dom recs
( F )  i^i 
{ dom recs ( F
) } )  =  (/)
128, 11eqtri 2303 . . . . 5  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/)
13 funun 5296 . . . . 5  |-  ( ( ( Fun recs ( F
)  /\  Fun  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  /\  ( dom recs ( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/) )  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
146, 12, 13sylancl 643 . . . 4  |-  ( dom recs
( F )  e.  On  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
157uneq2i 3326 . . . . 5  |-  ( dom recs
( F )  u. 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  u.  { dom recs ( F ) } )
16 dmun 4885 . . . . 5  |-  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  ( dom recs ( F )  u.  dom  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )
17 df-suc 4398 . . . . 5  |-  suc  dom recs ( F )  =  ( dom recs ( F )  u.  { dom recs ( F ) } )
1815, 16, 173eqtr4i 2313 . . . 4  |-  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F )
1914, 18jctir 524 . . 3  |-  ( dom recs
( F )  e.  On  ->  ( Fun  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  /\  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  =  suc  dom recs ( F ) ) )
20 df-fn 5258 . . 3  |-  ( (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  Fn 
suc  dom recs ( F )  <-> 
( Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  /\  dom  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  suc  dom recs ( F
) ) )
2119, 20sylibr 203 . 2  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
22 tfrlem.3 . . 3  |-  C  =  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
2322fneq1i 5338 . 2  |-  ( C  Fn  suc  dom recs ( F )  <->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
2421, 23sylibr 203 1  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643   Ord word 4391   Oncon0 4392   suc csuc 4394   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem11  6404  tfrlem12  6405  tfrlem13  6406
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-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-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-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  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-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-fv 5263  df-recs 6388
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