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Theorem frrlem9 8318
Description: Lemma for well-founded recursion. Show that the well-founded recursive generator produces a function. Hypothesis three will be eliminated using different induction rules depending on if we use partial orders or the axiom of infinity. (Contributed by Scott Fenton, 27-Aug-2022.)
Hypotheses
Ref Expression
frrlem9.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem9.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem9.3 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Assertion
Ref Expression
frrlem9 (𝜑 → Fun 𝐹)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝐵,𝑔,   𝑥,𝐹,𝑢,𝑣   𝜑,𝑓   𝑓,𝐹   𝜑,𝑔,,𝑥,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔,)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓)   𝑅(𝑣,𝑢,𝑔,)   𝐹(𝑦,𝑔,)   𝐺(𝑣,𝑢,𝑔,)

Proof of Theorem frrlem9
StepHypRef Expression
1 eluni2 4916 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐵 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
2 df-br 5149 . . . . . . . . 9 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐹)
3 frrlem9.1 . . . . . . . . . . 11 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
4 frrlem9.2 . . . . . . . . . . 11 𝐹 = frecs(𝑅, 𝐴, 𝐺)
53, 4frrlem5 8314 . . . . . . . . . 10 𝐹 = 𝐵
65eleq2i 2831 . . . . . . . . 9 (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
72, 6bitri 275 . . . . . . . 8 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
8 df-br 5149 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
98rexbii 3092 . . . . . . . 8 (∃𝑔𝐵 𝑥𝑔𝑢 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
101, 7, 93bitr4i 303 . . . . . . 7 (𝑥𝐹𝑢 ↔ ∃𝑔𝐵 𝑥𝑔𝑢)
11 eluni2 4916 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐵 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
12 df-br 5149 . . . . . . . . 9 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
135eleq2i 2831 . . . . . . . . 9 (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
1412, 13bitri 275 . . . . . . . 8 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
15 df-br 5149 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1615rexbii 3092 . . . . . . . 8 (∃𝐵 𝑥𝑣 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
1711, 14, 163bitr4i 303 . . . . . . 7 (𝑥𝐹𝑣 ↔ ∃𝐵 𝑥𝑣)
1810, 17anbi12i 628 . . . . . 6 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
19 reeanv 3227 . . . . . 6 (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
2018, 19bitr4i 278 . . . . 5 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ ∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣))
21 frrlem9.3 . . . . . 6 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2221rexlimdvva 3211 . . . . 5 (𝜑 → (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2320, 22biimtrid 242 . . . 4 (𝜑 → ((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2423alrimiv 1925 . . 3 (𝜑 → ∀𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2524alrimivv 1926 . 2 (𝜑 → ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
263, 4frrlem6 8315 . . 3 Rel 𝐹
27 dffun2 6573 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣)))
2826, 27mpbiran 709 . 2 (Fun 𝐹 ↔ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2925, 28sylibr 234 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  cop 4637   cuni 4912   class class class wbr 5148  cres 5691  Rel wrel 5694  Predcpred 6322  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431  frecscfrecs 8304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-frecs 8305
This theorem is referenced by:  frrlem10  8319  frrlem11  8320  frrlem12  8321  frrlem13  8322  fpr1  8327  fprfung  8333  frr1  9797
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