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Theorem tfrlem13 6422
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 6416 . . 3  |-  Ord  dom recs ( F )
3 ordirr 4426 . . 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 2296 . . . . 5  |-  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  =  (recs ( F )  u. 
{ <. dom recs ( F
) ,  ( F `
recs ( F ) ) >. } )
61, 5tfrlem12 6421 . . . 4  |-  (recs ( F )  e.  _V  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F
) ) >. } )  e.  A )
7 elssuni 3871 . . . . 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 6413 . . . . 5  |- recs ( F )  =  U. A
97, 8syl6sseqr 3238 . . . 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 4894 . . . 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 4955 . . . . . 6  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  _V )
14 elon2 4419 . . . . . 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 4486 . . . . 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 6419 . . . . 5  |-  ( dom recs
( F )  e.  On  ->  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } )  Fn  suc  dom recs
( F ) )
19 fndm 5359 . . . . 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 2373 . . 3  |-  (recs ( F )  e.  _V  ->  dom recs ( F )  e.  dom  (recs ( F )  u.  { <. dom recs ( F ) ,  ( F ` recs ( F ) ) >. } ) )
2211, 21sseldd 3194 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653   <.cop 3656   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271  recscrecs 6403
This theorem is referenced by:  tfrlem14  6423  tfrlem15  6424  tfrlem16  6425  tfr2b  6428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-recs 6404
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