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Theorem tfrlem14 7532
Description: Lemma for transfinite recursion. Assuming ax-rep 4804, dom recs ∈ V ↔ recs ∈ 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 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
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 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem13 7531 . . 3 ¬ recs(𝐹) ∈ V
31tfrlem7 7524 . . . 4 Fun recs(𝐹)
4 funex 6523 . . . 4 ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V)
53, 4mpan 706 . . 3 (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V)
62, 5mto 188 . 2 ¬ dom recs(𝐹) ∈ On
71tfrlem8 7525 . . 3 Ord dom recs(𝐹)
8 ordeleqon 7030 . . 3 (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On))
97, 8mpbi 220 . 2 (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)
106, 9mtpor 1735 1 dom recs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  dom cdm 5143  cres 5145  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  cfv 5926  recscrecs 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-wrecs 7452  df-recs 7513
This theorem is referenced by:  tfr1  7538
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