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

Theorem tfrlem14 6402
Description: Lemma for transfinite recursion. Assuming ax-rep 4132,  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 6401 . . 3  |-  -. recs ( F )  e.  _V
31tfrlem7 6394 . . . 4  |-  Fun recs ( F )
4 funex 5704 . . . 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 169 . 2  |-  -.  dom recs ( F )  e.  On
71tfrlem8 6395 . . . 4  |-  Ord  dom recs ( F )
8 ordeleqon 4579 . . . 4  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  e.  On  \/  dom recs ( F )  =  On ) )
97, 8mpbi 201 . . 3  |-  ( dom recs
( F )  e.  On  \/  dom recs ( F )  =  On )
109ori 366 . 2  |-  ( -. 
dom recs ( F )  e.  On  ->  dom recs ( F )  =  On )
116, 10ax-mp 10 1  |-  dom recs ( F )  =  On
Colors of variables: wff set class
Syntax hints:   -. wn 5    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   E.wrex 2545   _Vcvv 2789   Ord word 4390   Oncon0 4391   dom cdm 4688    |` cres 4690   Fun wfun 5215    Fn wfn 5216   ` cfv 5221  recscrecs 6382
This theorem is referenced by:  tfr1  6408
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383
  Copyright terms: Public domain W3C validator