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Theorem tfrlem14 6653
Description: Lemma for transfinite recursion. Assuming ax-rep 4321,  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 6652 . . 3  |-  -. recs ( F )  e.  _V
31tfrlem7 6645 . . . 4  |-  Fun recs ( F )
4 funex 5964 . . . 4  |-  ( ( Fun recs ( F )  /\  dom recs ( F
)  e.  On )  -> recs ( F )  e.  _V )
53, 4mpan 653 . . 3  |-  ( dom recs
( F )  e.  On  -> recs ( F
)  e.  _V )
62, 5mto 170 . 2  |-  -.  dom recs ( F )  e.  On
71tfrlem8 6646 . . 3  |-  Ord  dom recs ( F )
8 ordeleqon 4770 . . 3  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  e.  On  \/  dom recs ( F )  =  On ) )
97, 8mpbi 201 . 2  |-  ( dom recs
( F )  e.  On  \/  dom recs ( F )  =  On )
106, 9mtp-or 1548 1  |-  dom recs ( F )  =  On
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2423   A.wral 2706   E.wrex 2707   _Vcvv 2957   Ord word 4581   Oncon0 4582   dom cdm 4879    |` cres 4881   Fun wfun 5449    Fn wfn 5450   ` cfv 5455  recscrecs 6633
This theorem is referenced by:  tfr1  6659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-recs 6634
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