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Theorem frrlem9 8293
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 4907 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐵 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
2 df-br 5143 . . . . . . . . 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 8289 . . . . . . . . . 10 𝐹 = 𝐵
65eleq2i 2821 . . . . . . . . 9 (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
72, 6bitri 275 . . . . . . . 8 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
8 df-br 5143 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
98rexbii 3090 . . . . . . . 8 (∃𝑔𝐵 𝑥𝑔𝑢 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
101, 7, 93bitr4i 303 . . . . . . 7 (𝑥𝐹𝑢 ↔ ∃𝑔𝐵 𝑥𝑔𝑢)
11 eluni2 4907 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐵 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
12 df-br 5143 . . . . . . . . 9 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
135eleq2i 2821 . . . . . . . . 9 (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
1412, 13bitri 275 . . . . . . . 8 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
15 df-br 5143 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1615rexbii 3090 . . . . . . . 8 (∃𝐵 𝑥𝑣 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
1711, 14, 163bitr4i 303 . . . . . . 7 (𝑥𝐹𝑣 ↔ ∃𝐵 𝑥𝑣)
1810, 17anbi12i 627 . . . . . 6 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
19 reeanv 3222 . . . . . 6 (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
2018, 19bitr4i 278 . . . . 5 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ ∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣))
21 frrlem9.3 . . . . . 6 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2221rexlimdvva 3207 . . . . 5 (𝜑 → (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2320, 22biimtrid 241 . . . 4 (𝜑 → ((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2423alrimiv 1923 . . 3 (𝜑 → ∀𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2524alrimivv 1924 . 2 (𝜑 → ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
263, 4frrlem6 8290 . . 3 Rel 𝐹
27 dffun2 6552 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣)))
2826, 27mpbiran 708 . 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 1085  wal 1532   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wral 3057  wrex 3066  wss 3945  cop 4630   cuni 4903   class class class wbr 5142  cres 5674  Rel wrel 5677  Predcpred 6298  Fun wfun 6536   Fn wfn 6537  cfv 6542  (class class class)co 7414  frecscfrecs 8279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7417  df-frecs 8280
This theorem is referenced by:  frrlem10  8294  frrlem11  8295  frrlem12  8296  frrlem13  8297  fpr1  8302  fprfung  8308  frr1  9776
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