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Theorem tfrlem13 8004
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 7998 . . 3 Ord dom recs(𝐹)
3 ordirr 6185 . . 3 (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹))
42, 3ax-mp 5 . 2 ¬ dom recs(𝐹) ∈ dom recs(𝐹)
5 eqid 2820 . . . . 5 (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
61, 5tfrlem12 8003 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4844 . . . . 5 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ 𝐴)
81recsfval 7995 . . . . 5 recs(𝐹) = 𝐴
97, 8sseqtrrdi 3997 . . . 4 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10 dmss 5747 . . . 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 7591 . . . . . 6 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14 elon2 6178 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 585 . . . . 5 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
16 sucidg 6245 . . . . 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 8001 . . . . 5 (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 6431 . . . . 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 2914 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
2211, 21sseldd 3947 . 2 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹))
234, 22mto 199 1 ¬ recs(𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 398   = wceq 1537  wcel 2114  {cab 2798  wral 3125  wrex 3126  Vcvv 3473  cun 3911  wss 3913  {csn 4543  cop 4549   cuni 4814  dom cdm 5531  cres 5533  Ord word 6166  Oncon0 6167  suc csuc 6169   Fn wfn 6326  cfv 6331  recscrecs 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-fv 6339  df-wrecs 7925  df-recs 7986
This theorem is referenced by:  tfrlem14  8005  tfrlem15  8006  tfrlem16  8007  tfr2b  8010
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