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Theorem tfrlem13 7531
 Description: Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem13 ¬ recs(𝐹) ∈ V
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem13
StepHypRef Expression
1 tfrlem.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem8 7525 . . 3 Ord dom recs(𝐹)
3 ordirr 5779 . . 3 (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹))
42, 3ax-mp 5 . 2 ¬ dom recs(𝐹) ∈ dom recs(𝐹)
5 eqid 2651 . . . . 5 (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
61, 5tfrlem12 7530 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4499 . . . . 5 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ 𝐴)
81recsfval 7522 . . . . 5 recs(𝐹) = 𝐴
97, 8syl6sseqr 3685 . . . 4 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10 dmss 5355 . . . 4 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹) → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ dom recs(𝐹))
116, 9, 103syl 18 . . 3 (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ dom recs(𝐹))
122a1i 11 . . . . . 6 (recs(𝐹) ∈ V → Ord dom recs(𝐹))
13 dmexg 7139 . . . . . 6 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14 elon2 5772 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 699 . . . . 5 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
16 sucidg 5841 . . . . 5 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
1715, 16syl 17 . . . 4 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ suc dom recs(𝐹))
181, 5tfrlem10 7528 . . . . 5 (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 6028 . . . . 5 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹) → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹))
2015, 18, 193syl 18 . . . 4 (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹))
2117, 20eleqtrrd 2733 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
2211, 21sseldd 3637 . 2 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹))
234, 22mto 188 1 ¬ recs(𝐹) ∈ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210  ⟨cop 4216  ∪ cuni 4468  dom cdm 5143   ↾ cres 5145  Ord word 5760  Oncon0 5761  suc csuc 5763   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-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-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-fv 5934  df-wrecs 7452  df-recs 7513 This theorem is referenced by:  tfrlem14  7532  tfrlem15  7533  tfrlem16  7534  tfr2b  7537
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