MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem10 Unicode version

Theorem tfrlem10 6584
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 6588, 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 5682 . . . . . . 7  |-  ( F `
recs ( F ) )  e.  _V
2 funsng 5437 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  ( F ` recs ( F
) )  e.  _V )  ->  Fun  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
31, 2mpan2 653 . . . . . 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 6580 . . . . . 6  |-  Fun recs ( F )
63, 5jctil 524 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( Fun recs ( F )  /\  Fun  {
<. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
71dmsnop 5284 . . . . . . 7  |-  dom  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. }  =  { dom recs ( F ) }
87ineq2i 3482 . . . . . 6  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  i^i  { dom recs ( F ) } )
94tfrlem8 6581 . . . . . . 7  |-  Ord  dom recs ( F )
10 orddisj 4560 . . . . . . 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 2407 . . . . 5  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/)
13 funun 5435 . . . . 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 644 . . . 4  |-  ( dom recs
( F )  e.  On  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
157uneq2i 3441 . . . . 5  |-  ( dom recs
( F )  u. 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  u.  { dom recs ( F ) } )
16 dmun 5016 . . . . 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 4528 . . . . 5  |-  suc  dom recs ( F )  =  ( dom recs ( F )  u.  { dom recs ( F ) } )
1815, 16, 173eqtr4i 2417 . . . 4  |-  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F )
1914, 18jctir 525 . . 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 5397 . . 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 204 . 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 5479 . 2  |-  ( C  Fn  suc  dom recs ( F )  <->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
2421, 23sylibr 204 1  |-  ( dom recs
( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   E.wrex 2650   _Vcvv 2899    u. cun 3261    i^i cin 3262   (/)c0 3571   {csn 3757   <.cop 3760   Ord word 4521   Oncon0 4522   suc csuc 4524   dom cdm 4818    |` cres 4820   Fun wfun 5388    Fn wfn 5389   ` cfv 5394  recscrecs 6568
This theorem is referenced by:  tfrlem11  6585  tfrlem12  6586  tfrlem13  6587
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-recs 6569
  Copyright terms: Public domain W3C validator