Theorem frrlem10 8319

Description: Lemma for well-founded recursion. Under the compatibility hypothesis, compute the value of 𝐹 within its domain. (Contributed by Scott Fenton, 6-Dec-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
frrlem10	⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

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 frrlem10

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3482	. . . 4 ⊢ 𝑦 ∈ V
2	1	eldm2 5915	. . 3 ⊢ (𝑦 ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹)
3		frrlem9.1	. . . . . . . . 9 ⊢ 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	. . . . . . . . 9 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
5	3, 4	frrlem5 8314	. . . . . . . 8 ⊢ 𝐹 = ∪ 𝐵
6	3	unieqi 4924	. . . . . . . 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))))}
7	5, 6	eqtri 2763	. . . . . . 7 ⊢ 𝐹 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
8	7	eleq2i 2831	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
9		eluniab 4926	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10	8, 9	bitri 275	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
11		19.8a 2179	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
12	11	3ad2ant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
13		abid 2716	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
14	12, 13	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
15		elssuni 4942	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑓 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
16	14, 15	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑓 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
17	16, 7	sseqtrrdi 4047	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑓 ⊆ 𝐹)
18		simpl23 1252	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
19		simpl3 1192	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → ⟨𝑦, 𝑧⟩ ∈ 𝑓)
20		vex 3482	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
21	1, 20	opeldm 5921	. . . . . . . . . . . . . 14 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝑓 → 𝑦 ∈ dom 𝑓)
22	19, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → 𝑦 ∈ dom 𝑓)
23		simpl21 1250	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → 𝑓 Fn 𝑥)
24	23	fndmd 6674	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → dom 𝑓 = 𝑥)
25	22, 24	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → 𝑦 ∈ 𝑥)
26		rsp 3245	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦 ∈ 𝑥 → (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
27	18, 25, 26	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
28		simpl1 1190	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → 𝜑)
29		frrlem9.3	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
30	3, 4, 29	frrlem9 8318	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐹)
31	28, 30	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → Fun 𝐹)
32		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → 𝑓 ⊆ 𝐹)
33		funssfv 6928	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ 𝑦 ∈ dom 𝑓) → (𝐹‘𝑦) = (𝑓‘𝑦))
34	31, 32, 22, 33	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → (𝐹‘𝑦) = (𝑓‘𝑦))
35		simp22r 1292	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
36	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
37		rsp 3245	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
38	36, 25, 37	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
39	38, 24	sseqtrrd 4037	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓)
40		fun2ssres 6613	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
41	31, 32, 39, 40	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
42	41	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
43	27, 34, 42	3eqtr4d 2785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) ∧ 𝑓 ⊆ 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
44	17, 43	mpdan 687	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
45	44	3exp 1118	. . . . . . . 8 ⊢ (𝜑 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
46	45	exlimdv 1931	. . . . . . 7 ⊢ (𝜑 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
47	46	impcomd 411	. . . . . 6 ⊢ (𝜑 → ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
48	47	exlimdv 1931	. . . . 5 ⊢ (𝜑 → (∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
49	10, 48	biimtrid 242	. . . 4 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
50	49	exlimdv 1931	. . 3 ⊢ (𝜑 → (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
51	2, 50	biimtrid 242	. 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
52	51	imp 406	1 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  ∀wral 3059   ⊆ wss 3963  ⟨cop 4637  ∪ cuni 4912   class class class wbr 5148  dom cdm 5689   ↾ cres 5691  Predcpred 6322  Fun wfun 6557   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  ax-sep 5302  ax-nul 5312  ax-pr 5438

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-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  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-iun 4998  df-br 5149  df-opab 5211  df-id 5583  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  fpr2a  8326  frr2  9798