MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem14 Unicode version

Theorem tfrlem14 6449
Description: Lemma for transfinite recursion. Assuming ax-rep 4168,  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 6448 . . 3  |-  -. recs ( F )  e.  _V
31tfrlem7 6441 . . . 4  |-  Fun recs ( F )
4 funex 5784 . . . 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 6442 . . . 4  |-  Ord  dom recs ( F )
8 ordeleqon 4617 . . . 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 1633    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578   _Vcvv 2822   Ord word 4428   Oncon0 4429   dom cdm 4726    |` cres 4728   Fun wfun 5286    Fn wfn 5287   ` cfv 5292  recscrecs 6429
This theorem is referenced by:  tfr1  6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430
  Copyright terms: Public domain W3C validator