Theorem dffi3 9340

Description: The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypothesis

Ref	Expression
dffi3.1	⊢ 𝑅 = (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)))

Assertion

Ref	Expression
dffi3	⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉   𝑦,𝑢,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑢)   𝑅(𝑧,𝑢)   𝑉(𝑧,𝑢)

Proof of Theorem dffi3

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffi2 9332	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)})
2		fr0g 8365	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3		frfnom 8364	. . . . . . . . 9 ⊢ (rec(𝑅, 𝐴) ↾ ω) Fn ω
4		peano1 7829	. . . . . . . . 9 ⊢ ∅ ∈ ω
5		fnfvelrn 7018	. . . . . . . . 9 ⊢ (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
6	3, 4, 5	mp2an 692	. . . . . . . 8 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
7	2, 6	eqeltrrdi 2837	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8		elssuni 4891	. . . . . . 7 ⊢ (𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω) → 𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
10		reeanv 3201	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
11		eliun 4948	. . . . . . . . . 10 ⊢ (𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12		eliun 4948	. . . . . . . . . 10 ⊢ (𝑑 ∈ ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
13	11, 12	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
14		fniunfv 7187	. . . . . . . . . . . 12 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
15	14	eleq2d 2814	. . . . . . . . . . 11 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
16		fniunfv 7187	. . . . . . . . . . . 12 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
17	16	eleq2d 2814	. . . . . . . . . . 11 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 ∈ ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
18	15, 17	anbi12d 632	. . . . . . . . . 10 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ((𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))))
19	3, 18	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
20	10, 13, 19	3bitr2i 299	. . . . . . . 8 ⊢ (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
21		ordom 7816	. . . . . . . . . . . . . . . 16 ⊢ Ord ω
22		ordunel 7766	. . . . . . . . . . . . . . . 16 ⊢ ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚 ∪ 𝑛) ∈ ω)
23	21, 22	mp3an1 1450	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚 ∪ 𝑛) ∈ ω)
24	23	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚 ∪ 𝑛) ∈ ω)
25		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
26	24, 25	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27		nnon 7812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
28		nnon 7812	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
29	28	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
30		onsseleq 6352	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
31	27, 29, 30	syl2an2 686	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
32		rzal 4462	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
33	32	biantrud 531	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
34		fveq2 6826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
35	34	sseq1d 3969	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
36	33, 35	bitr3d 281	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ∅ → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
37		fveq2 6826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
38	37	sseq1d 3969	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
39	37	sseq2d 3970	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
40	39	raleqbi1dv 3302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
41	38, 40	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))))
42		fveq2 6826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
43	42	sseq1d 3969	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
44	42	sseq2d 3970	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
45	44	raleqbi1dv 3302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = suc 𝑛 → (∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
46	43, 45	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = suc 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
47		ssfii 9328	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
48	2, 47	eqsstrd 3972	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ 𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴))
49		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
50		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
51		ineq1 4166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎 = 𝑥 → (𝑎 ∩ 𝑏) = (𝑥 ∩ 𝑏))
52	51	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = 𝑥 → (𝑥 = (𝑎 ∩ 𝑏) ↔ 𝑥 = (𝑥 ∩ 𝑏)))
53		ineq2 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑏 = 𝑥 → (𝑥 ∩ 𝑏) = (𝑥 ∩ 𝑥))
54		inidm 4180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑥 ∩ 𝑥) = 𝑥
55	53, 54	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = 𝑥 → (𝑥 ∩ 𝑏) = 𝑥)
56	55	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑥 → (𝑥 = (𝑥 ∩ 𝑏) ↔ 𝑥 = 𝑥))
57	52, 56	rspc2ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 = 𝑥) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎 ∩ 𝑏))
58	49, 49, 50, 57	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎 ∩ 𝑏))
59		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏))
60	59	rnmpo 7486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎 ∩ 𝑏)}
61	60	eqabri 2871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎 ∩ 𝑏))
62	58, 61	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
63	62	ssriv 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏))
64		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
65		fvex 6839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
66	65	uniex 7681	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
67	66	pwex 5322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
68		inss1 4190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ∩ 𝑏) ⊆ 𝑎
69		elssuni 4891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
71	68, 70	sstrid 3949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
72		vex 3442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
73	72	inex1 5259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ∩ 𝑏) ∈ V
74	73	elpw 4557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
75	71, 74	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
76	75	rgen2 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
77	59	fmpo 8010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
78	76, 77	mpbi 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
79		frn 6663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ⊆ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
80	78, 79	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ⊆ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
81	67, 80	ssexi 5264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ∈ V
82		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ Ⅎ𝑣𝐴
83		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ Ⅎ𝑣𝑛
84		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ Ⅎ𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏))
85		dffi3.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 𝑅 = (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)))
86		mpoeq12 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑢 = 𝑣 ∧ 𝑢 = 𝑣) → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)))
87	86	anidms 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)))
88		ineq1 4166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 = 𝑎 → (𝑦 ∩ 𝑧) = (𝑎 ∩ 𝑧))
89		ineq2 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑏 → (𝑎 ∩ 𝑧) = (𝑎 ∩ 𝑏))
90	88, 89	cbvmpov 7448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)) = (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))
91	87, 90	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
92	91	rneqd 5884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑢 = 𝑣 → ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
93	92	cbvmptv 5199	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧))) = (𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
94	85, 93	eqtri 2752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ 𝑅 = (𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
95		rdgeq1 8340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑅 = (𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))) → rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))), 𝐴))
96	94, 95	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))), 𝐴)
97	96	reseq1i 5930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))), 𝐴) ↾ ω)
98		mpoeq12 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
99	98	anidms 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
100	99	rneqd 5884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
101	82, 83, 84, 97, 100	frsucmpt 8367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
102	64, 81, 101	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)))
103	63, 102	sseqtrrid 3981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
104		sstr2 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
105	103, 104	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
106	105	ralimdv 3143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
107		vex 3442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑛 ∈ V
108		fveq2 6826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
109	108	sseq1d 3969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
110	107, 109	ralsn 4635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
111	103, 110	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
112	106, 111	jctird 526	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
113		df-suc 6317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
114	113	raleqi 3288	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
115		ralunb 4150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
116	114, 115	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
117	112, 116	imbitrrdi 252	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
118		fiin 9331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎 ∩ 𝑏) ∈ (fi‘𝐴))
119	118	rgen2 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴)
120		ss2ralv 4008	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴)))
121	119, 120	mpi 20	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴))
122	59	fmpo 8010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
123	121, 122	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
124	123	frnd 6664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ⊆ (fi‘𝐴))
125	124	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ⊆ (fi‘𝐴))
126	102, 125	eqsstrd 3972	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
127	117, 126	jctild 525	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
128	127	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ω → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
129	128	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ω → (𝐴 ∈ 𝑉 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))))
130	36, 41, 46, 48, 129	finds2 7838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ω → (𝐴 ∈ 𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
131	130	impcom 407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
132	131	simprd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
133	132	r19.21bi 3221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
134	133	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝑦 ∈ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
135	134	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ∈ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136		fveq2 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137		eqimss 3996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
138	136, 137	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
139	138	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
140	135, 139	jaod 859	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
141	31, 140	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
142	141	ralrimiva 3121	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
143	142	ralrimiva 3121	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144	143	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
145		ssun1 4131	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
146	145	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚 ∪ 𝑛))
147		sseq2 3964	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑚 ∪ 𝑛)))
148		fveq2 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
149	148	sseq2d 3970	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
150	147, 149	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → ((𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (𝑦 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))))
151		sseq1 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚 ∪ 𝑛) ↔ 𝑚 ⊆ (𝑚 ∪ 𝑛)))
152		fveq2 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
153	152	sseq1d 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑚 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
154	151, 153	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑚 → ((𝑦 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))) ↔ (𝑚 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))))
155	150, 154	rspc2v 3590	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑚 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))))
156	26, 144, 146, 155	syl3c 66	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
157	156	sseld 3936	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
158		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
159	24, 158	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑛 ∈ ω))
160		ssun2 4132	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
161	160	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚 ∪ 𝑛))
162		sseq1 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚 ∪ 𝑛) ↔ 𝑛 ⊆ (𝑚 ∪ 𝑛)))
163	108	sseq1d 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
164	162, 163	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑛 → ((𝑦 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))) ↔ (𝑛 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))))
165	150, 164	rspc2v 3590	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑛 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑛 ⊆ (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))))
166	159, 144, 161, 165	syl3c 66	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
167	166	sseld 3936	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
168	23	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑚 ∪ 𝑛) ∈ ω)
169		peano2 7830	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∪ 𝑛) ∈ ω → suc (𝑚 ∪ 𝑛) ∈ ω)
170		fveq2 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = suc (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)))
171	170	ssiun2s 5000	. . . . . . . . . . . . . . 15 ⊢ (suc (𝑚 ∪ 𝑛) ∈ ω → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)) ⊆ ∪ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
172	168, 169, 171	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)) ⊆ ∪ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
173		simprl 770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
174		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
175		eqidd 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑑))
176		ineq1 4166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → (𝑎 ∩ 𝑏) = (𝑐 ∩ 𝑏))
177	176	eqeq2d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → ((𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑏)))
178		ineq2 4167	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑑 → (𝑐 ∩ 𝑏) = (𝑐 ∩ 𝑑))
179	178	eqeq2d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑑 → ((𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑑)))
180	177, 179	rspc2ev 3592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑑)) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏))
181	173, 174, 175, 180	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏))
182		vex 3442	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑐 ∈ V
183	182	inex1 5259	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∩ 𝑑) ∈ V
184		eqeq1 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑐 ∩ 𝑑) → (𝑥 = (𝑎 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏)))
185	184	2rexbidv 3194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑐 ∩ 𝑑) → (∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))𝑥 = (𝑎 ∩ 𝑏) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏)))
186	183, 185	elab 3637	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∩ 𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))𝑥 = (𝑎 ∩ 𝑏)} ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏))
187	181, 186	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))𝑥 = (𝑎 ∩ 𝑏)})
188		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏))
189	188	rnmpo 7486	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))𝑥 = (𝑎 ∩ 𝑏)}
190	187, 189	eleqtrrdi 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
191		fvex 6839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
192	191	uniex 7681	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
193	192	pwex 5322	. . . . . . . . . . . . . . . . 17 ⊢ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
194		elssuni 4891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
195	68, 194	sstrid 3949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
196	73	elpw 4557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↔ (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
197	195, 196	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → (𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
198	197	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))) → (𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
199	198	rgen2 3169	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))
200	188	fmpo 8010	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))(𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
201	199, 200	mpbi 230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))
202		frn 6663	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))⟶𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) ⊆ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
203	201, 202	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) ⊆ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))
204	193, 203	ssexi 5264	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) ∈ V
205		nfcv 2891	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑣(𝑚 ∪ 𝑛)
206		nfcv 2891	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏))
207		mpoeq12 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
208	207	anidms 566	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
209	208	rneqd 5884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
210	82, 205, 206, 97, 209	frsucmpt 8367	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∪ 𝑛) ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
211	168, 204, 210	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)))
212	190, 211	eleqtrrd 2831	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) ∈ ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)))
213	172, 212	sseldd 3938	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) ∈ ∪ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
214		fniunfv 7187	. . . . . . . . . . . . . 14 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ∪ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
215	3, 214	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ∪ ran (rec(𝑅, 𝐴) ↾ ω)
216	213, 215	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
217	216	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
218	157, 167, 217	syl2and 608	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
219	218	rexlimdvva 3186	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
220	219	imp 406	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
221	20, 220	sylan2br 595	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
222	221	ralrimivva 3172	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∀𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
223	131	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
224		fvex 6839	. . . . . . . . . . . . 13 ⊢ (fi‘𝐴) ∈ V
225	224	elpw2 5276	. . . . . . . . . . . 12 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
226	223, 225	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
227	226	ralrimiva 3121	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
228		fnfvrnss 7059	. . . . . . . . . 10 ⊢ (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴)) → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
229	3, 227, 228	sylancr 587	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
230		sspwuni 5052	. . . . . . . . 9 ⊢ (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
231	229, 230	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
232		ssexg 5265	. . . . . . . 8 ⊢ ((∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
233	231, 224, 232	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
234		sseq2 3964	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
235		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐 ∩ 𝑑) ∈ 𝑥 ↔ (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
236	235	raleqbi1dv 3302	. . . . . . . . . 10 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
237	236	raleqbi1dv 3302	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥 ↔ ∀𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
238	234, 237	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → ((𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥) ↔ (𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))))
239	238	elabg 3634	. . . . . . 7 ⊢ (∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ V → (∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)} ↔ (𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))))
240	233, 239	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)} ↔ (𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)(𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω))))
241	9, 222, 240	mpbir2and 713	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)})
242		intss1 4916	. . . . 5 ⊢ (∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)} ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
243	241, 242	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)} ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
244	1, 243	eqsstrd 3972	. . 3 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
245	244, 231	eqssd 3955	. 2 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
246		df-ima 5636	. . 3 ⊢ (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
247	246	unieqi 4873	. 2 ⊢ ∪ (rec(𝑅, 𝐴) “ ω) = ∪ ran (rec(𝑅, 𝐴) ↾ ω)
248	245, 247	eqtr4di 2782	1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861  ∩ cint 4899  ∪ ciun 4944   ↦ cmpt 5176   × cxp 5621  ran crn 5624   ↾ cres 5625   “ cima 5626  Ord word 6310  Oncon0 6311  suc csuc 6313   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486   ∈ cmpo 7355  ωcom 7806  reccrdg 8338  ficfi 9319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-en 8880  df-fin 8883  df-fi 9320

This theorem is referenced by: (None)