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Theorem tfrlem14 6638
Description: Lemma for transfinite recursion. Assuming ax-rep 4307,  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 6637 . . 3  |-  -. recs ( F )  e.  _V
31tfrlem7 6630 . . . 4  |-  Fun recs ( F )
4 funex 5949 . . . 4  |-  ( ( Fun recs ( F )  /\  dom recs ( F
)  e.  On )  -> recs ( F )  e.  _V )
53, 4mpan 652 . . 3  |-  ( dom recs
( F )  e.  On  -> recs ( F
)  e.  _V )
62, 5mto 169 . 2  |-  -.  dom recs ( F )  e.  On
71tfrlem8 6631 . . 3  |-  Ord  dom recs ( F )
8 ordeleqon 4755 . . 3  |-  ( Ord 
dom recs ( F )  <->  ( dom recs ( F )  e.  On  \/  dom recs ( F )  =  On ) )
97, 8mpbi 200 . 2  |-  ( dom recs
( F )  e.  On  \/  dom recs ( F )  =  On )
106, 9mtp-or 1547 1  |-  dom recs ( F )  =  On
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2416   A.wral 2692   E.wrex 2693   _Vcvv 2943   Ord word 4567   Oncon0 4568   dom cdm 4864    |` cres 4866   Fun wfun 5434    Fn wfn 5435   ` cfv 5440  recscrecs 6618
This theorem is referenced by:  tfr1  6644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-suc 4574  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-recs 6619
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