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Theorem tfrlem13 6339
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 6333 . . 3  |-  Ord  dom recs ( F )
3 ordirr 4347 . . 3  |-  ( Ord 
dom recs ( F )  ->  -.  dom recs ( F )  e.  dom recs ( F
) )
42, 3ax-mp 10 . 2  |-  -.  dom recs ( F )  e.  dom recs ( F )
5 eqid 2256 . . . . 5  |-  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )
61, 5tfrlem12 6338 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  A )
7 elssuni 3796 . . . . 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 6330 . . . . 5  |- recs ( F )  =  U. A
97, 8syl6sseqr 3167 . . . 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 4831 . . . 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 20 . . 3  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  C_  dom recs ( F ) )
122a1i 12 . . . . . 6  |-  (recs ( F )  e.  _V  ->  Ord  dom recs ( F
) )
13 dmexg 4892 . . . . . 6  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
14 elon2 4340 . . . . . 6  |-  ( dom recs
( F )  e.  On  <->  ( Ord  dom recs ( F )  /\  dom recs ( F )  e.  _V ) )
1512, 13, 14sylanbrc 648 . . . . 5  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  On )
16 sucidg 4407 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
1715, 16syl 17 . . . 4  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  suc  dom recs ( F ) )
181, 5tfrlem10 6336 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
19 fndm 5246 . . . . 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 20 . . . 4  |-  (recs ( F )  e.  _V  ->  dom  (recs ( F )  u.  { <. dom recs
( F ) ,  ( F ` recs ( F ) ) >. } )  =  suc  dom recs
( F ) )
2117, 20eleqtrrd 2333 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
2211, 21sseldd 3123 . 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 5    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516   E.wrex 2517   _Vcvv 2740    u. cun 3092    C_ wss 3094   {csn 3581   <.cop 3584   U.cuni 3768   Ord word 4328   Oncon0 4329   suc csuc 4331   dom cdm 4626    |` cres 4628    Fn wfn 4633   ` cfv 4638  recscrecs 6320
This theorem is referenced by:  tfrlem14  6340  tfrlem15  6341  tfrlem16  6342  tfr2b  6345
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-fv 4654  df-recs 6321
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