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Theorem frrlem9 33397
 Description: Lemma for founded recursion. Show that the founded recursive generator produces a function. Hypothesis three will be eliminated using different induction rules depending on if we use partial ordering or 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 4805 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐵 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
2 df-br 5036 . . . . . . . . 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 33393 . . . . . . . . . 10 𝐹 = 𝐵
65eleq2i 2843 . . . . . . . . 9 (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
72, 6bitri 278 . . . . . . . 8 (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐵)
8 df-br 5036 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
98rexbii 3175 . . . . . . . 8 (∃𝑔𝐵 𝑥𝑔𝑢 ↔ ∃𝑔𝐵𝑥, 𝑢⟩ ∈ 𝑔)
101, 7, 93bitr4i 306 . . . . . . 7 (𝑥𝐹𝑢 ↔ ∃𝑔𝐵 𝑥𝑔𝑢)
11 eluni2 4805 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐵 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
12 df-br 5036 . . . . . . . . 9 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
135eleq2i 2843 . . . . . . . . 9 (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
1412, 13bitri 278 . . . . . . . 8 (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐵)
15 df-br 5036 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1615rexbii 3175 . . . . . . . 8 (∃𝐵 𝑥𝑣 ↔ ∃𝐵𝑥, 𝑣⟩ ∈ )
1711, 14, 163bitr4i 306 . . . . . . 7 (𝑥𝐹𝑣 ↔ ∃𝐵 𝑥𝑣)
1810, 17anbi12i 629 . . . . . 6 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
19 reeanv 3285 . . . . . 6 (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) ↔ (∃𝑔𝐵 𝑥𝑔𝑢 ∧ ∃𝐵 𝑥𝑣))
2018, 19bitr4i 281 . . . . 5 ((𝑥𝐹𝑢𝑥𝐹𝑣) ↔ ∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣))
21 frrlem9.3 . . . . . 6 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2221rexlimdvva 3218 . . . . 5 (𝜑 → (∃𝑔𝐵𝐵 (𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
2320, 22syl5bi 245 . . . 4 (𝜑 → ((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2423alrimiv 1928 . . 3 (𝜑 → ∀𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2524alrimivv 1929 . 2 (𝜑 → ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
263, 4frrlem6 33394 . . 3 Rel 𝐹
27 dffun2 6349 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣)))
2826, 27mpbiran 708 . 2 (Fun 𝐹 ↔ ∀𝑥𝑢𝑣((𝑥𝐹𝑢𝑥𝐹𝑣) → 𝑢 = 𝑣))
2925, 28sylibr 237 1 (𝜑 → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860  ⟨cop 4531  ∪ cuni 4801   class class class wbr 5035   ↾ cres 5529  Rel wrel 5532  Predcpred 6129  Fun wfun 6333   Fn wfn 6334  ‘cfv 6339  (class class class)co 7155  frecscfrecs 33383 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-iota 6298  df-fun 6341  df-fn 6342  df-fv 6347  df-ov 7158  df-frecs 33384 This theorem is referenced by:  frrlem10  33398  frrlem11  33399  frrlem12  33400  frrlem13  33401  fpr1  33405  frr1  33410
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