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Theorem tfrlem14 7641
Description: Lemma for transfinite recursion. Assuming ax-rep 4905, 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 7640 . . 3 ¬ recs(𝐹) ∈ V
31tfrlem7 7633 . . . 4 Fun recs(𝐹)
4 funex 6627 . . . 4 ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V)
53, 4mpan 664 . . 3 (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V)
62, 5mto 188 . 2 ¬ dom recs(𝐹) ∈ On
71tfrlem8 7634 . . 3 Ord dom recs(𝐹)
8 ordeleqon 7136 . . 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 1843 1 dom recs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wa 382  wo 828   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  dom cdm 5250  cres 5252  Ord word 5866  Oncon0 5867  Fun wfun 6026   Fn wfn 6027  cfv 6032  recscrecs 7621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-wrecs 7560  df-recs 7622
This theorem is referenced by:  tfr1  7647
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