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Theorem dffi3 8895

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 8887	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)})
2		fr0g 8071	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3		frfnom 8070	. . . . . . . . 9 ⊢ (rec(𝑅, 𝐴) ↾ ω) Fn ω
4		peano1 7601	. . . . . . . . 9 ⊢ ∅ ∈ ω
5		fnfvelrn 6848	. . . . . . . . 9 ⊢ (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
6	3, 4, 5	mp2an 690	. . . . . . . 8 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
7	2, 6	eqeltrrdi 2922	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8		elssuni 4868	. . . . . . 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 3367	. . . . . . . . 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 4923	. . . . . . . . . 10 ⊢ (𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12		eliun 4923	. . . . . . . . . 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 7006	. . . . . . . . . . . 12 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
15	14	eleq2d 2898	. . . . . . . . . . 11 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 ∈ ∪ 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
16		fniunfv 7006	. . . . . . . . . . . 12 ⊢ ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ∪ 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
17	16	eleq2d 2898	. . . . . . . . . . 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 301	. . . . . . . 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 7589	. . . . . . . . . . . . . . . 16 ⊢ Ord ω
22		ordunel 7542	. . . . . . . . . . . . . . . 16 ⊢ ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚 ∪ 𝑛) ∈ ω)
23	21, 22	mp3an1 1444	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚 ∪ 𝑛) ∈ ω)
24	23	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚 ∪ 𝑛) ∈ ω)
25		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
26	24, 25	jca 514	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27		nnon 7586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
28		nnon 7586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
29	28	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
30		onsseleq 6232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
31	27, 29, 30	syl2an2 684	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
32		rzal 4453	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
33	32	biantrud 534	. . . . . . . . . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
35	34	sseq1d 3998	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
36	33, 35	bitr3d 283	. . . . . . . . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
38	37	sseq1d 3998	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
39	37	sseq2d 3999	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3403	. . . . . . . . . . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
43	42	sseq1d 3998	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
44	42	sseq2d 3999	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3403	. . . . . . . . . . . . . . . . . . . . . . . . 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 8883	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
48	2, 47	eqsstrd 4005	. . . . . . . . . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
51		ineq1 4181	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎 = 𝑥 → (𝑎 ∩ 𝑏) = (𝑥 ∩ 𝑏))
52	51	eqeq2d 2832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑎 = 𝑥 → (𝑥 = (𝑎 ∩ 𝑏) ↔ 𝑥 = (𝑥 ∩ 𝑏)))
53		ineq2 4183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑏 = 𝑥 → (𝑥 ∩ 𝑏) = (𝑥 ∩ 𝑥))
54		inidm 4195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑥 ∩ 𝑥) = 𝑥
55	53, 54	syl6eq 2872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑏 = 𝑥 → (𝑥 ∩ 𝑏) = 𝑥)
56	55	eqeq2d 2832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑏 = 𝑥 → (𝑥 = (𝑥 ∩ 𝑏) ↔ 𝑥 = 𝑥))
57	52, 56	rspc2ev 3635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎 ∩ 𝑏))
59		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	abeq2i 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏))
64		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
65		fvex 6683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
66	65	uniex 7467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
67	66	pwex 5281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
68		inss1 4205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ∩ 𝑏) ⊆ 𝑎
69		elssuni 4868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
70	69	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
71	68, 70	sstrid 3978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
72		vex 3497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
73	72	inex1 5221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ∩ 𝑏) ∈ V
74	73	elpw 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
75	71, 74	sylibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
76	75	rgen2 3203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
77	59	fmpo 7766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎 ∩ 𝑏)) ∈ V
82		nfcv 2977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ Ⅎ𝑣𝐴
83		nfcv 2977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ Ⅎ𝑣𝑛
84		nfcv 2977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑢 = 𝑣 ∧ 𝑢 = 𝑣) → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)))
87	86	anidms 569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)))
88		ineq1 4181	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 = 𝑎 → (𝑦 ∩ 𝑧) = (𝑎 ∩ 𝑧))
89		ineq2 4183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑏 → (𝑎 ∩ 𝑧) = (𝑎 ∩ 𝑏))
90	88, 89	cbvmpov 7249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑦 ∈ 𝑣, 𝑧 ∈ 𝑣 ↦ (𝑦 ∩ 𝑧)) = (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))
91	87, 90	syl6eq 2872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑢 = 𝑣 → (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
92	91	rneqd 5808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑢 = 𝑣 → ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)) = ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
93	92	cbvmptv 5169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧))) = (𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
94	85, 93	eqtri 2844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ 𝑅 = (𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏)))
95		rdgeq1 8047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ (𝑎 ∩ 𝑏))), 𝐴) ↾ ω)
98		mpoeq12 7227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
104		sstr2 3974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑛 ∈ V
108		fveq2 6670	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
109	108	sseq1d 3998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4619	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
114	113	raleqi 3413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl6ibr 254	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 8886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎 ∩ 𝑏) ∈ (fi‘𝐴))
119	118	rgen2 3203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎 ∩ 𝑏) ∈ (fi‘𝐴)
120		ss2ralv 4035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
127	117, 126	jctild 528	. . . . . . . . . . . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . . . . . . . . . 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 7610	. . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ 𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
133	132	r19.21bi 3208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
134	133	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝑦 ∈ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
135	134	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ∈ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136		fveq2 6670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137		eqimss 4023	. . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
142	141	ralrimiva 3182	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
143	142	ralrimiva 3182	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦 ⊆ 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144	143	adantr 483	. . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
146	145	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚 ∪ 𝑛))
147		sseq2 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑚 ∪ 𝑛)))
148		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
149	148	sseq2d 3999	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚 ∪ 𝑛) ↔ 𝑚 ⊆ (𝑚 ∪ 𝑛)))
152		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
153	152	sseq1d 3998	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 3633	. . . . . . . . . . . . 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 3966	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
158		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
159	24, 158	jca 514	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑛 ∈ ω))
160		ssun2 4149	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
161	160	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚 ∪ 𝑛))
162		sseq1 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚 ∪ 𝑛) ↔ 𝑛 ⊆ (𝑚 ∪ 𝑛)))
163	108	sseq1d 3998	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 3633	. . . . . . . . . . . . 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 3966	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛))))
168	23	ad2antlr 725	. . . . . . . . . . . . . . 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 7602	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∪ 𝑛) ∈ ω → suc (𝑚 ∪ 𝑛) ∈ ω)
170		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = suc (𝑚 ∪ 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚 ∪ 𝑛)))
171	170	ssiun2s 4972	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . 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 4181	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → (𝑎 ∩ 𝑏) = (𝑐 ∩ 𝑏))
177	176	eqeq2d 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → ((𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑏)))
178		ineq2 4183	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑑 → (𝑐 ∩ 𝑏) = (𝑐 ∩ 𝑑))
179	178	eqeq2d 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑑 → ((𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑐 ∩ 𝑑)))
180	177, 179	rspc2ev 3635	. . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . 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 3497	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑐 ∈ V
183	182	inex1 5221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∩ 𝑑) ∈ V
184		eqeq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑐 ∩ 𝑑) → (𝑥 = (𝑎 ∩ 𝑏) ↔ (𝑐 ∩ 𝑑) = (𝑎 ∩ 𝑏)))
185	184	2rexbidv 3300	. . . . . . . . . . . . . . . . . 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 3667	. . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . 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 7284	. . . . . . . . . . . . . . . 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 2924	. . . . . . . . . . . . . . 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 6683	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
192	191	uniex 7467	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
193	192	pwex 5281	. . . . . . . . . . . . . . . . 17 ⊢ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ∈ V
194		elssuni 4868	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → 𝑎 ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
195	68, 194	sstrid 3978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
196	73	elpw 4543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↔ (𝑎 ∩ 𝑏) ⊆ ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
197	195, 196	sylibr 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) → (𝑎 ∩ 𝑏) ∈ 𝒫 ∪ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)))
198	197	adantr 483	. . . . . . . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . . . . . 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 7766	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 6520	. . . . . . . . . . . . . . . . . 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 5226	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏)) ∈ V
205		nfcv 2977	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑣(𝑚 ∪ 𝑛)
206		nfcv 2977	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚 ∪ 𝑛)) ↦ (𝑎 ∩ 𝑏))
207		mpoeq12 7227	. . . . . . . . . . . . . . . . . . 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 569	. . . . . . . . . . . . . . . . . 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 5808	. . . . . . . . . . . . . . . . 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 8073	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 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 2916	. . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . 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 7006	. . . . . . . . . . . . . 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 2923	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 609	. . . . . . . . . 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 3294	. . . . . . . . 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 409	. . . . . . . 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 596	. . . . . . 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 3191	. . . . . 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 497	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
224		fvex 6683	. . . . . . . . . . . . 13 ⊢ (fi‘𝐴) ∈ V
225	224	elpw2 5248	. . . . . . . . . . . 12 ⊢ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
226	223, 225	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
227	226	ralrimiva 3182	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
228		fnfvrnss 6884	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
230		sspwuni 5022	. . . . . . . . 9 ⊢ (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
231	229, 230	sylib 220	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
232		ssexg 5227	. . . . . . . 8 ⊢ ((∪ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
233	231, 224, 232	sylancl 588	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
234		sseq2 3993	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
235		eleq2 2901	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐 ∩ 𝑑) ∈ 𝑥 ↔ (𝑐 ∩ 𝑑) ∈ ∪ ran (rec(𝑅, 𝐴) ↾ ω)))
236	235	raleqbi1dv 3403	. . . . . . . . . 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 3403	. . . . . . . . 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 3666	. . . . . . 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 711	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ∀𝑐 ∈ 𝑥 ∀𝑑 ∈ 𝑥 (𝑐 ∩ 𝑑) ∈ 𝑥)})
242		intss1 4891	. . . . 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 4005	. . 3 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) ⊆ ∪ ran (rec(𝑅, 𝐴) ↾ ω))
245	244, 231	eqssd 3984	. 2 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ ran (rec(𝑅, 𝐴) ↾ ω))
246		df-ima 5568	. . 3 ⊢ (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
247	246	unieqi 4851	. 2 ⊢ ∪ (rec(𝑅, 𝐴) “ ω) = ∪ ran (rec(𝑅, 𝐴) ↾ ω)
248	245, 247	syl6eqr 2874	1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ∩ cint 4876 ∪ ciun 4919 ↦ cmpt 5146 × cxp 5553 ran crn 5556 ↾ cres 5557 “ cima 5558 Ord word 6190 Oncon0 6191 suc csuc 6193 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 ∈ cmpo 7158 ωcom 7580 reccrdg 8045 ficfi 8874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875

This theorem is referenced by: (None)

W3C validator