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Theorem tfrlem13 8419
Description: Lemma for transfinite recursion. If recs is a set function, then 𝐢 is acceptable, and thus a subset of recs, but dom 𝐢 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 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
Assertion
Ref Expression
tfrlem13 ¬ recs(𝐹) ∈ V
Distinct variable group:   π‘₯,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑓)

Proof of Theorem tfrlem13
StepHypRef Expression
1 tfrlem.1 . . . 4 𝐴 = {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
21tfrlem8 8413 . . 3 Ord dom recs(𝐹)
3 ordirr 6392 . . 3 (Ord dom recs(𝐹) β†’ Β¬ dom recs(𝐹) ∈ dom recs(𝐹))
42, 3ax-mp 5 . 2 Β¬ dom recs(𝐹) ∈ dom recs(𝐹)
5 eqid 2728 . . . . 5 (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) = (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
61, 5tfrlem12 8418 . . . 4 (recs(𝐹) ∈ V β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4944 . . . . 5 ((recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ 𝐴 β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) βŠ† βˆͺ 𝐴)
81recsfval 8410 . . . . 5 recs(𝐹) = βˆͺ 𝐴
97, 8sseqtrrdi 4033 . . . 4 ((recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ 𝐴 β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) βŠ† recs(𝐹))
10 dmss 5909 . . . 4 ((recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) βŠ† recs(𝐹) β†’ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) βŠ† dom recs(𝐹))
116, 9, 103syl 18 . . 3 (recs(𝐹) ∈ V β†’ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) βŠ† dom recs(𝐹))
122a1i 11 . . . . . 6 (recs(𝐹) ∈ V β†’ Ord dom recs(𝐹))
13 dmexg 7917 . . . . . 6 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ V)
14 elon2 6385 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 581 . . . . 5 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ On)
16 sucidg 6455 . . . . 5 (dom recs(𝐹) ∈ On β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
1715, 16syl 17 . . . 4 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
181, 5tfrlem10 8416 . . . . 5 (dom recs(𝐹) ∈ On β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 6662 . . . . 5 ((recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) Fn suc dom recs(𝐹) β†’ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) = suc dom recs(𝐹))
2015, 18, 193syl 18 . . . 4 (recs(𝐹) ∈ V β†’ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) = suc dom recs(𝐹))
2117, 20eleqtrrd 2832 . . 3 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}))
2211, 21sseldd 3983 . 2 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ dom recs(𝐹))
234, 22mto 196 1 Β¬ recs(𝐹) ∈ V
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  {csn 4632  βŸ¨cop 4638  βˆͺ cuni 4912  dom cdm 5682   β†Ύ cres 5684  Ord word 6373  Oncon0 6374  suc csuc 6376   Fn wfn 6548  β€˜cfv 6553  recscrecs 8399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7429  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400
This theorem is referenced by:  tfrlem14  8420  tfrlem15  8421  tfrlem16  8422  tfr2b  8425
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