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Theorem tfrlem13 6406
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 6400 . . 3  |-  Ord  dom recs ( F )
3 ordirr 4410 . . 3  |-  ( Ord 
dom recs ( F )  ->  -.  dom recs ( F )  e.  dom recs ( F
) )
42, 3ax-mp 8 . 2  |-  -.  dom recs ( F )  e.  dom recs ( F )
5 eqid 2283 . . . . 5  |-  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )
61, 5tfrlem12 6405 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  A )
7 elssuni 3855 . . . . 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 6397 . . . . 5  |- recs ( F )  =  U. A
97, 8syl6sseqr 3225 . . . 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 4878 . . . 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 18 . . 3  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  C_  dom recs ( F ) )
122a1i 10 . . . . . 6  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
13 dmexg 4939 . . . . . 6  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
14 elon2 4403 . . . . . 6  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
1512, 13, 14sylanbrc 645 . . . . 5  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
16 sucidg 4470 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
1715, 16syl 15 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
181, 5tfrlem10 6403 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
19 fndm 5343 . . . . 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 18 . . . 4  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F ) )
2117, 20eleqtrrd 2360 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
2211, 21sseldd 3181 . 2  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom recs ( F
) )
234, 22mto 167 1  |-  -. recs ( F )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640   <.cop 3643   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfrlem14  6407  tfrlem15  6408  tfrlem16  6409  tfr2b  6412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-recs 6388
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