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Theorem tfrlem13 8430
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 8424 . . 3 Ord dom recs(𝐹)
3 ordirr 6402 . . 3 (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹))
42, 3ax-mp 5 . 2 ¬ dom recs(𝐹) ∈ dom recs(𝐹)
5 eqid 2737 . . . . 5 (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
61, 5tfrlem12 8429 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4937 . . . . 5 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ 𝐴)
81recsfval 8421 . . . . 5 recs(𝐹) = 𝐴
97, 8sseqtrrdi 4025 . . . 4 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10 dmss 5913 . . . 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 7923 . . . . . 6 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14 elon2 6395 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 583 . . . . 5 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
16 sucidg 6465 . . . . 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 8427 . . . . 5 (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 6671 . . . . 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 2844 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
2211, 21sseldd 3984 . 2 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹))
234, 22mto 197 1 ¬ recs(𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cun 3949  wss 3951  {csn 4626  cop 4632   cuni 4907  dom cdm 5685  cres 5687  Ord word 6383  Oncon0 6384  suc csuc 6386   Fn wfn 6556  cfv 6561  recscrecs 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  tfrlem14  8431  tfrlem15  8432  tfrlem16  8433  tfr2b  8436
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