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Theorem tfrlem13 6610
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 6604 . . 3  |-  Ord  dom recs ( F )
3 ordirr 4559 . . 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 2404 . . . . 5  |-  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )
61, 5tfrlem12 6609 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  A )
7 elssuni 4003 . . . . 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 6601 . . . . 5  |- recs ( F )  =  U. A
97, 8syl6sseqr 3355 . . . 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 5028 . . . 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 5089 . . . . . 6  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
14 elon2 4552 . . . . . 6  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
1512, 13, 14sylanbrc 646 . . . . 5  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
16 sucidg 4619 . . . . 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 6607 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
19 fndm 5503 . . . . 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 2481 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
2211, 21sseldd 3309 . 2  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom recs ( F
) )
234, 22mto 169 1  |-  -. recs ( F )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    u. cun 3278    C_ wss 3280   {csn 3774   <.cop 3777   U.cuni 3975   Ord word 4540   Oncon0 4541   suc csuc 4543   dom cdm 4837    |` cres 4839    Fn wfn 5408   ` cfv 5413  recscrecs 6591
This theorem is referenced by:  tfrlem14  6611  tfrlem15  6612  tfrlem16  6613  tfr2b  6616
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-recs 6592
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