Theorem tfrlem14 8320

Description: Lemma for transfinite recursion. Assuming ax-rep 5199, 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 ∧ ∀𝑦 ∈ 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

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem13 8319	. . 3 ⊢ ¬ recs(𝐹) ∈ V
3	1	tfrlem7 8312	. . . 4 ⊢ Fun recs(𝐹)
4		funex 7163	. . . 4 ⊢ ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V)
5	3, 4	mpan 696	. . 3 ⊢ (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V)
6	2, 5	mto 198	. 2 ⊢ ¬ dom recs(𝐹) ∈ On
7	1	tfrlem8 8313	. . 3 ⊢ Ord dom recs(𝐹)
8		ordeleqon 7725	. . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On))
9	7, 8	mpbi 231	. 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)
10	6, 9	mtpor 1777	1 ⊢ dom recs(𝐹) = On

Syntax hints:   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431  dom cdm 5618   ↾ cres 5620  Ord word 6309  Oncon0 6310  Fun wfun 6479   Fn wfn 6480  ‘cfv 6485  recscrecs 8300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301

This theorem is referenced by:  tfr1  8326