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Theorem tfrlem13 8391
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 8385 . . 3 Ord dom recs(𝐹)
3 ordirr 6376 . . 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 2726 . . . . 5 (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) = (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
61, 5tfrlem12 8390 . . . 4 (recs(𝐹) ∈ V β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ 𝐴)
7 elssuni 4934 . . . . 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 8382 . . . . 5 recs(𝐹) = βˆͺ 𝐴
97, 8sseqtrrdi 4028 . . . 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 5896 . . . 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 7891 . . . . . 6 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ V)
14 elon2 6369 . . . . . 6 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
1512, 13, 14sylanbrc 582 . . . . 5 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ On)
16 sucidg 6439 . . . . 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 8388 . . . . 5 (dom recs(𝐹) ∈ On β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19 fndm 6646 . . . . 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 2830 . . 3 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ dom (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}))
2211, 21sseldd 3978 . 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 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  {csn 4623  βŸ¨cop 4629  βˆͺ cuni 4902  dom cdm 5669   β†Ύ cres 5671  Ord word 6357  Oncon0 6358  suc csuc 6360   Fn wfn 6532  β€˜cfv 6537  recscrecs 8371
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372
This theorem is referenced by:  tfrlem14  8392  tfrlem15  8393  tfrlem16  8394  tfr2b  8397
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