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

Theorem tfrlem10 6677
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 6681, 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 5771 . . . . . . 7  |-  ( F `
recs ( F ) )  e.  _V
2 funsng 5526 . . . . . . 7  |-  ( ( dom recs ( F )  e.  On  /\  ( F ` recs ( F
) )  e.  _V )  ->  Fun  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )
31, 2mpan2 654 . . . . . 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 6673 . . . . . 6  |-  Fun recs ( F )
63, 5jctil 525 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  ( Fun recs ( F )  /\  Fun  {
<. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
71dmsnop 5373 . . . . . . 7  |-  dom  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. }  =  { dom recs ( F ) }
87ineq2i 3525 . . . . . 6  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  i^i  { dom recs ( F ) } )
94tfrlem8 6674 . . . . . . 7  |-  Ord  dom recs ( F )
10 orddisj 4648 . . . . . . 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 2462 . . . . 5  |-  ( dom recs
( F )  i^i 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  (/)
13 funun 5524 . . . . 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 645 . . . 4  |-  ( dom recs
( F )  e.  On  ->  Fun  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
157uneq2i 3484 . . . . 5  |-  ( dom recs
( F )  u. 
dom  { <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  =  ( dom recs ( F
)  u.  { dom recs ( F ) } )
16 dmun 5105 . . . . 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 4616 . . . . 5  |-  suc  dom recs ( F )  =  ( dom recs ( F )  u.  { dom recs ( F ) } )
1815, 16, 173eqtr4i 2472 . . . 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 5486 . . 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 5568 . 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 4    /\ wa 360    = wceq 1653    e. wcel 1727   {cab 2428   A.wral 2711   E.wrex 2712   _Vcvv 2962    u. cun 3304    i^i cin 3305   (/)c0 3613   {csn 3838   <.cop 3841   Ord word 4609   Oncon0 4610   suc csuc 4612   dom cdm 4907    |` cres 4909   Fun wfun 5477    Fn wfn 5478   ` cfv 5483  recscrecs 6661
This theorem is referenced by:  tfrlem11  6678  tfrlem12  6679  tfrlem13  6680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491  df-recs 6662
  Copyright terms: Public domain W3C validator