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Theorem frrlem9 8275
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 4904 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐵 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
2 df-br 5140 . . . . . . . . 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 8271 . . . . . . . . . 10 𝐹 = 𝐵
65eleq2i 2817 . . . . . . . . 9 (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
72, 6bitri 275 . . . . . . . 8 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
8 df-br 5140 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
98rexbii 3086 . . . . . . . 8 (∃𝑔𝐵 𝑥𝑔𝑢 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
101, 7, 93bitr4i 303 . . . . . . 7 (𝑥𝐹𝑢 ↔ ∃𝑔𝐵 𝑥𝑔𝑢)
11 eluni2 4904 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐵 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
12 df-br 5140 . . . . . . . . 9 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
135eleq2i 2817 . . . . . . . . 9 (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
1412, 13bitri 275 . . . . . . . 8 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
15 df-br 5140 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1615rexbii 3086 . . . . . . . 8 (∃𝐵 𝑥𝑣 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
1711, 14, 163bitr4i 303 . . . . . . 7 (𝑥𝐹𝑣 ↔ ∃𝐵 𝑥𝑣)
1810, 17anbi12i 626 . . . . . 6 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
19 reeanv 3218 . . . . . 6 (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
2018, 19bitr4i 278 . . . . 5 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ ∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣))
21 frrlem9.3 . . . . . 6 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2221rexlimdvva 3203 . . . . 5 (𝜑 → (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2320, 22biimtrid 241 . . . 4 (𝜑 → ((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2423alrimiv 1922 . . 3 (𝜑 → ∀𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2524alrimivv 1923 . 2 (𝜑 → ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
263, 4frrlem6 8272 . . 3 Rel 𝐹
27 dffun2 6544 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣)))
2826, 27mpbiran 706 . 2 (Fun 𝐹 ↔ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2925, 28sylibr 233 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wral 3053  wrex 3062  wss 3941  cop 4627   cuni 4900   class class class wbr 5139  cres 5669  Rel wrel 5672  Predcpred 6290  Fun wfun 6528   Fn wfn 6529  cfv 6534  (class class class)co 7402  frecscfrecs 8261
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-ov 7405  df-frecs 8262
This theorem is referenced by:  frrlem10  8276  frrlem11  8277  frrlem12  8278  frrlem13  8279  fpr1  8284  fprfung  8290  frr1  9751
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