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Theorem tfrlem13 6680
Description: Lemma for transfinite recursion. If recs is a set function, then  C is acceptable, and thus a subset of recs, but 
dom  C is bigger than  dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem13  |-  -. recs ( F )  e.  _V
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem13
StepHypRef Expression
1 tfrlem.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem8 6674 . . 3  |-  Ord  dom recs ( F )
3 ordirr 4628 . . 3  |-  ( Ord 
dom recs ( F )  ->  -.  dom recs ( F )  e.  dom recs ( F
) )
42, 3ax-mp 5 . 2  |-  -.  dom recs ( F )  e.  dom recs ( F )
5 eqid 2442 . . . . 5  |-  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )
61, 5tfrlem12 6679 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  A )
7 elssuni 4067 . . . . 5  |-  ( (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  e.  A  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  C_  U. A
)
81recsfval 6671 . . . . 5  |- recs ( F )  =  U. A
97, 8syl6sseqr 3381 . . . 4  |-  ( (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  e.  A  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  C_ recs ( F ) )
10 dmss 5098 . . . 4  |-  ( (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  C_ recs ( F )  ->  dom  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  C_  dom recs ( F ) )
116, 9, 103syl 19 . . 3  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  C_  dom recs ( F ) )
122a1i 11 . . . . . 6  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
13 dmexg 5159 . . . . . 6  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
14 elon2 4621 . . . . . 6  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
1512, 13, 14sylanbrc 647 . . . . 5  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
16 sucidg 4688 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
1715, 16syl 16 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
181, 5tfrlem10 6677 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
19 fndm 5573 . . . . 5  |-  ( (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )  Fn 
suc  dom recs ( F )  ->  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F ) )
2015, 18, 193syl 19 . . . 4  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F ) )
2117, 20eleqtrrd 2519 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
2211, 21sseldd 3335 . 2  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom recs ( F
) )
234, 22mto 170 1  |-  -. recs ( F )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1727   {cab 2428   A.wral 2711   E.wrex 2712   _Vcvv 2962    u. cun 3304    C_ wss 3306   {csn 3838   <.cop 3841   U.cuni 4039   Ord word 4609   Oncon0 4610   suc csuc 4612   dom cdm 4907    |` cres 4909    Fn wfn 5478   ` cfv 5483  recscrecs 6661
This theorem is referenced by:  tfrlem14  6681  tfrlem15  6682  tfrlem16  6683  tfr2b  6686
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
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