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Theorem tfrlem10 7054
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 7058, 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 5862 . . . . . . 7  |-  ( F `
recs ( F ) )  e.  _V
2 funsng 5620 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  ( F ` recs ( F
) )  e.  _V )  ->  Fun  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
31, 2mpan2 671 . . . . . 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 7050 . . . . . 6  |-  Fun recs ( F )
63, 5jctil 537 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( Fun recs ( F )  /\  Fun  {
<. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
71dmsnop 5468 . . . . . . 7  |-  dom  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. }  =  { dom recs ( F ) }
87ineq2i 3679 . . . . . 6  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  i^i  { dom recs ( F ) } )
94tfrlem8 7051 . . . . . . 7  |-  Ord  dom recs ( F )
10 orddisj 4902 . . . . . . 7  |-  ( Ord 
dom recs ( F )  -> 
( dom recs ( F
)  i^i  { dom recs ( F ) } )  =  (/) )
119, 10ax-mp 5 . . . . . 6  |-  ( dom recs
( F )  i^i 
{ dom recs ( F
) } )  =  (/)
128, 11eqtri 2470 . . . . 5  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/)
13 funun 5616 . . . . 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 662 . . . 4  |-  ( dom recs
( F )  e.  On  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
157uneq2i 3637 . . . . 5  |-  ( dom recs
( F )  u. 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  u.  { dom recs ( F ) } )
16 dmun 5195 . . . . 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 4870 . . . . 5  |-  suc  dom recs ( F )  =  ( dom recs ( F )  u.  { dom recs ( F ) } )
1815, 16, 173eqtr4i 2480 . . . 4  |-  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F )
1914, 18jctir 538 . . 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 5577 . . 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 212 . 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 5661 . 2  |-  ( C  Fn  suc  dom recs ( F )  <->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
2421, 23sylibr 212 1  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   {cab 2426   A.wral 2791   E.wrex 2792   _Vcvv 3093    u. cun 3456    i^i cin 3457   (/)c0 3767   {csn 4010   <.cop 4016   Ord word 4863   Oncon0 4864   suc csuc 4866   dom cdm 4985    |` cres 4987   Fun wfun 5568    Fn wfn 5569   ` cfv 5574  recscrecs 7039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-recs 7040
This theorem is referenced by:  tfrlem11  7055  tfrlem12  7056  tfrlem13  7057
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