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Theorem frrlem9 8229
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 4873 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐵 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
2 df-br 5110 . . . . . . . . 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 8225 . . . . . . . . . 10 𝐹 = 𝐵
65eleq2i 2826 . . . . . . . . 9 (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
72, 6bitri 275 . . . . . . . 8 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
8 df-br 5110 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
98rexbii 3094 . . . . . . . 8 (∃𝑔𝐵 𝑥𝑔𝑢 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
101, 7, 93bitr4i 303 . . . . . . 7 (𝑥𝐹𝑢 ↔ ∃𝑔𝐵 𝑥𝑔𝑢)
11 eluni2 4873 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐵 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
12 df-br 5110 . . . . . . . . 9 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
135eleq2i 2826 . . . . . . . . 9 (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
1412, 13bitri 275 . . . . . . . 8 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
15 df-br 5110 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1615rexbii 3094 . . . . . . . 8 (∃𝐵 𝑥𝑣 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
1711, 14, 163bitr4i 303 . . . . . . 7 (𝑥𝐹𝑣 ↔ ∃𝐵 𝑥𝑣)
1810, 17anbi12i 628 . . . . . 6 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
19 reeanv 3216 . . . . . 6 (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
2018, 19bitr4i 278 . . . . 5 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ ∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣))
21 frrlem9.3 . . . . . 6 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2221rexlimdvva 3202 . . . . 5 (𝜑 → (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2320, 22biimtrid 241 . . . 4 (𝜑 → ((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2423alrimiv 1931 . . 3 (𝜑 → ∀𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2524alrimivv 1932 . 2 (𝜑 → ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
263, 4frrlem6 8226 . . 3 Rel 𝐹
27 dffun2 6510 . . 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 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3914  cop 4596   cuni 4869   class class class wbr 5109  cres 5639  Rel wrel 5642  Predcpred 6256  Fun wfun 6494   Fn wfn 6495  cfv 6500  (class class class)co 7361  frecscfrecs 8215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7364  df-frecs 8216
This theorem is referenced by:  frrlem10  8230  frrlem11  8231  frrlem12  8232  frrlem13  8233  fpr1  8238  fprfung  8244  frr1  9703
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