Theorem frrlem8 8296

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 3461	. . 3 ⊢ 𝑧 ∈ V
2	1	eldm2 5893	. 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 8293	. . . . . . 7 ⊢ 𝐹 = ∪ 𝐵
6	3	frrlem1 8289	. . . . . . . 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 4886	. . . . . . 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 2788	. . . . . 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 2857	. . . . 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 4888	. . . . 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 278	. . . 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 1252	. . . . . . . . . . 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 3461	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
14	1, 13	opeldm 5899	. . . . . . . . . . . . 13 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝑔 → 𝑧 ∈ dom 𝑔)
15	14	adantr 486	. . . . . . . . . . . 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 1213	. . . . . . . . . . . . 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 6644	. . . . . . . . . . . 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 2867	. . . . . . . . . . 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 3255	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎 → (𝑧 ∈ 𝑎 → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑎))
20	12, 18, 19	sylc 66	. . . . . . . . . 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 3975	. . . . . . . . 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 2220	. . . . . . . . . . . . . 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 2907	. . . . . . . . . . . . . 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 237	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 4906	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝐵 → 𝑔 ⊆ ∪ 𝐵)
27	25, 26	syl 18	. . . . . . . . . . 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 3979	. . . . . . . . . 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 5894	. . . . . . . . . 10 ⊢ (𝑔 ⊆ 𝐹 → dom 𝑔 ⊆ dom 𝐹)
30	28, 29	syl 18	. . . . . . . . 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 3948	. . . . . . . 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 419	. . . . . . 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 1963	. . . . . 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 413	. . . . 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 1963	. . . 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 220	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
37	36	exlimiv 1963	. 2 ⊢ (∃𝑤⟨𝑧, 𝑤⟩ ∈ 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
38	2, 37	sylbi 220	1 ⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∀wral 3081   ⊆ wss 3906  ⟨cop 4597  ∪ cuni 4874  dom cdm 5663   ↾ cres 5665  Predcpred 6305   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419  frecscfrecs 8283

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

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-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-ov 7422  df-frecs 8284

This theorem is used by:  frrlem12  8300  frrlem13  8301  frrdmcl  8311