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 ∧ (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 4907	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
2		df-br 5143	. . . . . . . . 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 8289	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
6	5	eleq2i 2821	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
7	2, 6	bitri 275	. . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
8		df-br 5143	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
9	8	rexbii 3090	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 ⟨𝑥, 𝑢⟩ ∈ 𝑔)
10	1, 7, 9	3bitr4i 303	. . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢)
11		eluni2 4907	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
12		df-br 5143	. . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐹)
13	5	eleq2i 2821	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
14	12, 13	bitri 275	. . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
15		df-br 5143	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
16	15	rexbii 3090	. . . . . . . 8 ⊢ (∃ℎ ∈ 𝐵 𝑥ℎ𝑣 ↔ ∃ℎ ∈ 𝐵 ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	11, 14, 16	3bitr4i 303	. . . . . . 7 ⊢ (𝑥𝐹𝑣 ↔ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)
18	10, 17	anbi12i 627	. . . . . 6 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣))
19		reeanv 3222	. . . . . 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 3207	. . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
23	20, 22	biimtrid 241	. . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
24	23	alrimiv 1923	. . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
25	24	alrimivv 1924	. 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
26	3, 4	frrlem6 8290	. . 3 ⊢ Rel 𝐹
27		dffun2 6552	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)))
28	26, 27	mpbiran 708	. 2 ⊢ (Fun 𝐹 ↔ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))
29	25, 28	sylibr 233	1 ⊢ (𝜑 → Fun 𝐹)

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