Theorem frrlem9 8294

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 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem9.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem9.3	⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 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,ℎ   𝑥,𝐹,𝑢,𝑣   𝜑,𝑓   𝑓,𝐹   𝜑,𝑔,ℎ,𝑥,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣, 𝑢, 𝑔, ℎ)   𝐵(𝑥, 𝑦, 𝑣, 𝑢, 𝑓)   𝑅(𝑣, 𝑢, 𝑔, ℎ)   𝐹(𝑦, 𝑔, ℎ)   𝐺(𝑣, 𝑢, 𝑔, ℎ)

Proof of Theorem frrlem9

Step	Hyp	Ref	Expression
1		eluni2 4871	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
2		df-br 5104	. . . . . . . . 9 ⊢ (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐹)
3		frrlem9.1	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
4		frrlem9.2	. . . . . . . . . . 11 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
5	3, 4	frrlem5 8290	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
6	5	eleq2i 2852	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
7	2, 6	bitri 278	. . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
8		df-br 5104	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
9	8	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
10	1, 7, 9	3bitr4i 306	. . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢)
11		eluni2 4871	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
12		df-br 5104	. . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
13	5	eleq2i 2852	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
14	12, 13	bitri 278	. . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
15		df-br 5104	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
16	15	rexbii 3109	. . . . . . . 8 ⊢ (∃ℎ ∈ 𝐵 𝑥ℎ𝑣 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	11, 14, 16	3bitr4i 306	. . . . . . 7 ⊢ (𝑥𝐹𝑣 ↔ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)
18	10, 17	anbi12i 640	. . . . . 6 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣))
19		reeanv 3234	. . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣))
20	18, 19	bitr4i 281	. . . . 5 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣))
21		frrlem9.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
22	21	rexlimdvva 3219	. . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
23	20, 22	biimtrid 245	. . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
24	23	alrimiv 1960	. . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
25	24	alrimivv 1961	. 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
26	3, 4	frrlem6 8291	. . 3 ⊢ Rel 𝐹
27		dffun2 6543	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)))
28	26, 27	mpbiran 722	. 2 ⊢ (Fun 𝐹 ↔ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
29	25, 28	sylibr 237	1 ⊢ (𝜑 → Fun 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5103   ↾ cres 5657  Rel wrel 5660  Predcpred 6298  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414  frecscfrecs 8280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ov 7417  df-frecs 8281

This theorem is used by:  frrlem10  8295  frrlem11  8296  frrlem12  8297  frrlem13  8298  fpr1  8303  fprfung  8309  frr1  9744