Theorem frrlem9 8238

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 4855	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
2		df-br 5087	. . . . . . . . 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 8234	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
6	5	eleq2i 2829	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
7	2, 6	bitri 275	. . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
8		df-br 5087	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
9	8	rexbii 3085	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
10	1, 7, 9	3bitr4i 303	. . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢)
11		eluni2 4855	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
12		df-br 5087	. . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
13	5	eleq2i 2829	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
14	12, 13	bitri 275	. . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
15		df-br 5087	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
16	15	rexbii 3085	. . . . . . . 8 ⊢ (∃ℎ ∈ 𝐵 𝑥ℎ𝑣 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	11, 14, 16	3bitr4i 303	. . . . . . 7 ⊢ (𝑥𝐹𝑣 ↔ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)
18	10, 17	anbi12i 629	. . . . . 6 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣))
19		reeanv 3210	. . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣))
20	18, 19	bitr4i 278	. . . . 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 3195	. . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
23	20, 22	biimtrid 242	. . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
24	23	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
25	24	alrimivv 1930	. 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
26	3, 4	frrlem6 8235	. . 3 ⊢ Rel 𝐹
27		dffun2 6503	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)))
28	26, 27	mpbiran 710	. 2 ⊢ (Fun 𝐹 ↔ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
29	25, 28	sylibr 234	1 ⊢ (𝜑 → Fun 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ⟨cop 4574  ∪ cuni 4851   class class class wbr 5086   ↾ cres 5627  Rel wrel 5630  Predcpred 6259  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  frecscfrecs 8224

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ov 7364  df-frecs 8225

This theorem is referenced by:  frrlem10  8239  frrlem11  8240  frrlem12  8241  frrlem13  8242  fpr1  8247  fprfung  8253  frr1  9677