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Theorem tfrlem10 6371
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 6375, 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 5472 . . . . . . 7  |-  ( F `
recs ( F ) )  e.  _V
2 funsng 5236 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  ( F ` recs ( F
) )  e.  _V )  ->  Fun  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
31, 2mpan2 655 . . . . . 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 6367 . . . . . 6  |-  Fun recs ( F )
63, 5jctil 525 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( Fun recs ( F )  /\  Fun  {
<. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
71dmsnop 5134 . . . . . . 7  |-  dom  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. }  =  { dom recs ( F ) }
87ineq2i 3342 . . . . . 6  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  i^i  { dom recs ( F ) } )
94tfrlem8 6368 . . . . . . 7  |-  Ord  dom recs ( F )
10 orddisj 4402 . . . . . . 7  |-  ( Ord 
dom recs ( F )  -> 
( dom recs ( F
)  i^i  { dom recs ( F ) } )  =  (/) )
119, 10ax-mp 10 . . . . . 6  |-  ( dom recs
( F )  i^i 
{ dom recs ( F
) } )  =  (/)
128, 11eqtri 2278 . . . . 5  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/)
13 funun 5234 . . . . 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 646 . . . 4  |-  ( dom recs
( F )  e.  On  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
157uneq2i 3301 . . . . 5  |-  ( dom recs
( F )  u. 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  u.  { dom recs ( F ) } )
16 dmun 4873 . . . . 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 4370 . . . . 5  |-  suc  dom recs ( F )  =  ( dom recs ( F )  u.  { dom recs ( F ) } )
1815, 16, 173eqtr4i 2288 . . . 4  |-  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F )
1914, 18jctir 526 . . 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 4684 . . 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 205 . 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 5276 . 2  |-  ( C  Fn  suc  dom recs ( F )  <->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
2421, 23sylibr 205 1  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2244   A.wral 2518   E.wrex 2519   _Vcvv 2763    u. cun 3125    i^i cin 3126   (/)c0 3430   {csn 3614   <.cop 3617   Ord word 4363   Oncon0 4364   suc csuc 4366   dom cdm 4661    |` cres 4663   Fun wfun 4667    Fn wfn 4668   ` cfv 4673  recscrecs 6355
This theorem is referenced by:  tfrlem11  6372  tfrlem12  6373  tfrlem13  6374
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689  df-recs 6356
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