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Theorem tfrlem14 6407
Description: Lemma for transfinite recursion. Assuming ax-rep 4131,  dom recs  e.  _V  <-> recs  e. 
_V, so since  dom recs is an ordinal, it must be equal to  On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
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
tfrlem14  |-  dom recs ( F )  =  On
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem14
StepHypRef Expression
1 tfrlem.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem13 6406 . . 3  |-  -. recs ( F )  e.  _V
31tfrlem7 6399 . . . 4  |-  Fun recs ( F )
4 funex 5743 . . . 4  |-  ( ( Fun recs ( F )  /\  dom recs ( F
)  e.  On )  -> recs ( F )  e.  _V )
53, 4mpan 651 . . 3  |-  ( dom recs
( F )  e.  On  -> recs ( F
)  e.  _V )
62, 5mto 167 . 2  |-  -.  dom recs ( F )  e.  On
71tfrlem8 6400 . . . 4  |-  Ord  dom recs ( F )
8 ordeleqon 4580 . . . 4  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  e.  On  \/  dom recs ( F )  =  On ) )
97, 8mpbi 199 . . 3  |-  ( dom recs
( F )  e.  On  \/  dom recs ( F )  =  On )
109ori 364 . 2  |-  ( -. 
dom recs ( F )  e.  On  ->  dom recs ( F )  =  On )
116, 10ax-mp 8 1  |-  dom recs ( F )  =  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   Ord word 4391   Oncon0 4392   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  tfr1  6413
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-rep 4131  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-reu 2550  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-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388
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