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Theorem tfrlem10 7766
Description: Lemma for transfinite recursion. We define class 𝐶 by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about 𝐶 that will lead to a contradiction in tfrlem14 7770, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that 𝐶 is a function extending the domain of recs(𝐹) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem.3 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
Assertion
Ref Expression
tfrlem10 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐶   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem10
StepHypRef Expression
1 fvex 6459 . . . . . 6 (𝐹‘recs(𝐹)) ∈ V
2 funsng 6185 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
31, 2mpan2 681 . . . . 5 (dom recs(𝐹) ∈ On → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4 tfrlem.1 . . . . . 6 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
54tfrlem7 7762 . . . . 5 Fun recs(𝐹)
63, 5jctil 515 . . . 4 (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
71dmsnop 5863 . . . . . 6 dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} = {dom recs(𝐹)}
87ineq2i 4033 . . . . 5 (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∩ {dom recs(𝐹)})
94tfrlem8 7763 . . . . . 6 Ord dom recs(𝐹)
10 orddisj 6014 . . . . . 6 (Ord dom recs(𝐹) → (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅)
119, 10ax-mp 5 . . . . 5 (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅
128, 11eqtri 2801 . . . 4 (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅
13 funun 6180 . . . 4 (((Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∧ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅) → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
146, 12, 13sylancl 580 . . 3 (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
157uneq2i 3986 . . . 4 (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
16 dmun 5576 . . . 4 dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
17 df-suc 5982 . . . 4 suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
1815, 16, 173eqtr4i 2811 . . 3 dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)
19 df-fn 6138 . . 3 ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹) ↔ (Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∧ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)))
2014, 18, 19sylanblrc 584 . 2 (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
21 tfrlem.3 . . 3 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
2221fneq1i 6230 . 2 (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
2320, 22sylibr 226 1 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  {cab 2762  wral 3089  wrex 3090  Vcvv 3397  cun 3789  cin 3790  c0 4140  {csn 4397  cop 4403  dom cdm 5355  cres 5357  Ord word 5975  Oncon0 5976  suc csuc 5978  Fun wfun 6129   Fn wfn 6130  cfv 6135  recscrecs 7750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-wrecs 7689  df-recs 7751
This theorem is referenced by:  tfrlem11  7767  tfrlem12  7768  tfrlem13  7769
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