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Theorem tfrlem13 7432
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 7426 . . 3 Ord dom recs(𝐹)
3 ordirr 5703 . . 3 (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹))
42, 3ax-mp 5 . 2 ¬ dom recs(𝐹) ∈ dom recs(𝐹)
5 eqid 2626 . . . . 5 (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
61, 5tfrlem12 7431 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4438 . . . . 5 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ 𝐴)
81recsfval 7423 . . . . 5 recs(𝐹) = 𝐴
97, 8syl6sseqr 3636 . . . 4 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10 dmss 5288 . . . 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 7045 . . . . . 6 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14 elon2 5696 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 697 . . . . 5 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
16 sucidg 5765 . . . . 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 7429 . . . . 5 (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 5950 . . . . 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 2707 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
2211, 21sseldd 3589 . 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 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191  cun 3558  wss 3560  {csn 4153  cop 4159   cuni 4407  dom cdm 5079  cres 5081  Ord word 5684  Oncon0 5685  suc csuc 5687   Fn wfn 5845  cfv 5850  recscrecs 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-wrecs 7353  df-recs 7414
This theorem is referenced by:  tfrlem14  7433  tfrlem15  7434  tfrlem16  7435  tfr2b  7438
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