Theorem frrlem8 8317

Description: Lemma for well-founded recursion. dom 𝐹 is closed under predecessor classes. (Contributed by Scott Fenton, 6-Dec-2022.)

Hypotheses

Ref	Expression
frrlem5.1	⊢ 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))))}
frrlem5.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
frrlem8	⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑦,𝐹   𝑧,𝐴,𝑓,𝑥,𝑦   𝑧,𝑅   𝑧,𝐺

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑓)   𝐹(𝑥,𝑧,𝑓)

Proof of Theorem frrlem8

Dummy variables 𝑔 𝑎 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3482	. . 3 ⊢ 𝑧 ∈ V
2	1	eldm2 5915	. 2 ⊢ (𝑧 ∈ dom 𝐹 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ 𝐹)
3		frrlem5.1	. . . . . . . 8 ⊢ 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		frrlem5.2	. . . . . . . 8 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
5	3, 4	frrlem5 8314	. . . . . . 7 ⊢ 𝐹 = ∪ 𝐵
6	3	frrlem1 8310	. . . . . . . 8 ⊢ 𝐵 = {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
7	6	unieqi 4924	. . . . . . 7 ⊢ ∪ 𝐵 = ∪ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
8	5, 7	eqtri 2763	. . . . . 6 ⊢ 𝐹 = ∪ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))}
9	8	eleq2i 2831	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ∪ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))})
10		eluniab 4926	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ ∪ {𝑔 ∣ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))} ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))))
11	9, 10	bitri 275	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝐹 ↔ ∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))))
12		simpr2r 1232	. . . . . . . . . . 11 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎)
13		vex 3482	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
14	1, 13	opeldm 5921	. . . . . . . . . . . . 13 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 → 𝑧 ∈ dom 𝑔)
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧 ∈ dom 𝑔)
16		simpr1 1193	. . . . . . . . . . . . 13 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 Fn 𝑎)
17	16	fndmd 6674	. . . . . . . . . . . 12 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 = 𝑎)
18	15, 17	eleqtrd 2841	. . . . . . . . . . 11 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑧 ∈ 𝑎)
19		rsp 3245	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎 → (𝑧 ∈ 𝑎 → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎))
20	12, 18, 19	sylc 65	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎)
21	20, 17	sseqtrrd 4037	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝑔)
22		19.8a 2179	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
23	6	eqabri 2883	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))))
24	22, 23	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝑔 ∈ 𝐵)
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ∈ 𝐵)
26		elssuni 4942	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝐵 → 𝑔 ⊆ ∪ 𝐵)
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ⊆ ∪ 𝐵)
28	27, 5	sseqtrrdi 4047	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝑔 ⊆ 𝐹)
29		dmss 5916	. . . . . . . . . 10 ⊢ (𝑔 ⊆ 𝐹 → dom 𝑔 ⊆ dom 𝐹)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → dom 𝑔 ⊆ dom 𝐹)
31	21, 30	sstrd 4006	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ (𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
32	31	expcom 413	. . . . . . 7 ⊢ ((𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹))
33	32	exlimiv 1928	. . . . . 6 ⊢ (∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (⟨𝑧, 𝑤⟩ ∈ 𝑔 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹))
34	33	impcom 407	. . . . 5 ⊢ ((⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
35	34	exlimiv 1928	. . . 4 ⊢ (∃𝑔(⟨𝑧, 𝑤⟩ ∈ 𝑔 ∧ ∃𝑎(𝑔 Fn 𝑎 ∧ (𝑎 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎) ∧ ∀𝑧 ∈ 𝑎 (𝑔‘𝑧) = (𝑧𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑧))))) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
36	11, 35	sylbi 217	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
37	36	exlimiv 1928	. 2 ⊢ (∃𝑤⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
38	2, 37	sylbi 217	1 ⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  ∀wral 3059   ⊆ wss 3963  ⟨cop 4637  ∪ cuni 4912  dom cdm 5689   ↾ cres 5691  Predcpred 6322   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  frecscfrecs 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-frecs 8305

This theorem is referenced by:  frrlem12  8321  frrlem13  8322  frrdmcl  8332