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Theorem tfrlem14 7186
Description: Lemma for transfinite recursion. Assuming ax-rep 4548, 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 7185 . . 3 ¬ recs(𝐹) ∈ V
31tfrlem7 7178 . . . 4 Fun recs(𝐹)
4 funex 6201 . . . 4 ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V)
53, 4mpan 693 . . 3 (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V)
62, 5mto 183 . 2 ¬ dom recs(𝐹) ∈ On
71tfrlem8 7179 . . 3 Ord dom recs(𝐹)
8 ordeleqon 6692 . . 3 (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On))
97, 8mpbi 215 . 2 (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)
106, 9mtpor 1683 1 dom recs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wo 377  wa 378   = wceq 1468  wcel 1937  {cab 2491  wral 2791  wrex 2792  Vcvv 3066  dom cdm 4880  cres 4882  Ord word 5473  Oncon0 5474  Fun wfun 5627   Fn wfn 5628  cfv 5633  recscrecs 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-wrecs 7105  df-recs 7167
This theorem is referenced by:  tfr1  7192
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