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Theorem tfrlem7 6182
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem7 Fun recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem7
Dummy variables 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem6 6181 . 2 Rel recs(𝐹)
31recsfval 6180 . . . . . . . . 9 recs(𝐹) = 𝐴
43eleq2i 2184 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐴)
5 eluni 3709 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
64, 5bitri 183 . . . . . . 7 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
73eleq2i 2184 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐴)
8 eluni 3709 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐴 ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
97, 8bitri 183 . . . . . . 7 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
106, 9anbi12i 455 . . . . . 6 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
11 eeanv 1884 . . . . . 6 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
1210, 11bitr4i 186 . . . . 5 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)))
13 df-br 3900 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14 df-br 3900 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1513, 14anbi12i 455 . . . . . . . 8 ((𝑥𝑔𝑢𝑥𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ))
161tfrlem5 6179 . . . . . . . . 9 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1716impcom 124 . . . . . . . 8 (((𝑥𝑔𝑢𝑥𝑣) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1815, 17sylanbr 283 . . . . . . 7 (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1918an4s 562 . . . . . 6 (((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2019exlimivv 1852 . . . . 5 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2112, 20sylbi 120 . . . 4 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2221ax-gen 1410 . . 3 𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2322gen2 1411 . 2 𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24 dffun4 5104 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)))
252, 23, 24mpbir2an 911 1 Fun recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wral 2393  wrex 2394  cop 3500   cuni 3706   class class class wbr 3899  Oncon0 4255  cres 4511  Rel wrel 4514  Fun wfun 5087   Fn wfn 5088  cfv 5093  recscrecs 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfrlem9  6184  tfrfun  6185  tfrlemibfn  6193  tfrlemiubacc  6195  tfri1d  6200  rdgfun  6238
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