Theorem rdgssun 38444

Description: In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022.)

Hypotheses

Ref	Expression
rdgssun.1	⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))
rdgssun.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
rdgssun	⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))

Distinct variable groups:   𝑤,𝐴   𝑤,𝑌

Allowed substitution hints:   𝐵(𝑤)   𝐹(𝑤)   𝑋(𝑤)

Proof of Theorem rdgssun

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsbc1v 3762	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
2		0ex 5264	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
3		rzal 4453	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
4		sbceq1a 3753	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ [∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
5	3, 4	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → [∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
6	1, 2, 5	vtoclef 3528	. . . . . . . . . . 11 ⊢ [∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
7		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
8	7	elsuc 6442	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ suc 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
9		ssun1 4179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (rec(𝐹, 𝐴)‘𝑥) ⊆ ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
10		fvex 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (rec(𝐹, 𝐴)‘𝑥) ∈ V
11		rdgssun.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐵 ∈ V
12	11	csbex 5268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵 ∈ V
13	10, 12	unex 7771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵) ∈ V
14		nfcv 2926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤𝐴
15		nfcv 2926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤𝑥
16		rdgssun.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))
17		nfmpt1 5207	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑤(𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))
18	16, 17	nfcxfr 2924	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑤𝐹
19	18, 14	nfrdg 8436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤rec(𝐹, 𝐴)
20	19, 15	nffv 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤(rec(𝐹, 𝐴)‘𝑥)
21	20	nfcsb1 3874	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵
22	20, 21	nfun 4172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
23		rdgeq1 8433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵)) → rec(𝐹, 𝐴) = rec((𝑤 ∈ V ↦ (𝑤 ∪ 𝐵)), 𝐴))
24	16, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ rec(𝐹, 𝐴) = rec((𝑤 ∈ V ↦ (𝑤 ∪ 𝐵)), 𝐴)
25		id 23	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝑤 = (rec(𝐹, 𝐴)‘𝑥))
26		csbeq1a 3865	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝐵 = ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
27	25, 26	uneq12d 4171	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → (𝑤 ∪ 𝐵) = ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵))
28	14, 15, 22, 24, 27	rdgsucmptf 8450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵) ∈ V) → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵))
29	13, 28	mpan2 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵))
30	9, 29	sseqtrrid 3956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
31		sstr2 3920	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → ((rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
32	30, 31	syl5com 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
33	32	imim2d 58	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))))
34	33	imp 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
35		fveq2 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑥))
36	35	sseq1d 3944	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥) ↔ (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
37	30, 36	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
38	37	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
39	34, 38	jaod 873	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → ((𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
40	8, 39	biimtrid 245	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) → (𝑦 ∈ suc 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
41	40	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) → (𝑦 ∈ suc 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))))
42	41	ralimdv2 3175	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
43		df-sbc 3743	. . . . . . . . . . . . 13 ⊢ ([suc 𝑥 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ suc 𝑥 ∈ {𝑥 ∣ ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)})
44		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
45	44	sucex 7830	. . . . . . . . . . . . . 14 ⊢ suc 𝑥 ∈ V
46		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘suc 𝑥))
47	46	sseq2d 3945	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
48	47	raleqbi1dv 3333	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑥 → (∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
49		fveq2 6896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑧))
50	49	sseq2d 3945	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
51	50	raleqbi1dv 3333	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
52	51	cbvabv 2834	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} = {𝑧 ∣ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)}
53	45, 48, 52	elab2 3639	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ {𝑥 ∣ ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} ↔ ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
54	43, 53	bitri 278	. . . . . . . . . . . 12 ⊢ ([suc 𝑥 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑥(rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
55	42, 54	imbitrrdi 255	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → [suc 𝑥 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
56		ssiun2 5009	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑧 → (rec(𝐹, 𝐴)‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
57	56	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
58		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
59		rdglim2a 8455	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
60	58, 59	mpan 703	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑧 → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
61	60	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
62	57, 61	sseqtrrd 3950	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
63	62	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ (Lim 𝑧 → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
64		df-sbc 3743	. . . . . . . . . . . . . . 15 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑥 ∣ ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)})
65	52	eleq2i 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑥 ∣ ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)} ↔ 𝑧 ∈ {𝑧 ∣ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)})
66	64, 65	bitri 278	. . . . . . . . . . . . . 14 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ 𝑧 ∈ {𝑧 ∣ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)})
67		abid 2745	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ {𝑧 ∣ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)} ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
68	66, 67	bitri 278	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
69	63, 68	sylibr 237	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
70	69	a1d 26	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
71	6, 55, 70	tfindes 7884	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
72		rsp 3254	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
73	71, 72	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
74		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ On ↔ 𝑋 ∈ On))
75	74	adantl 487	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑥 ∈ On ↔ 𝑋 ∈ On))
76		eleq12 2854	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝑥 ↔ 𝑌 ∈ 𝑋))
77		fveq2 6896	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
78	77	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
79		fveq2 6896	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
80	79	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
81	78, 80	sseq12d 3946	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
82	76, 81	imbi12d 347	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → ((𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)) ↔ (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
83	75, 82	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ On → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))) ↔ (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
84	73, 83	mpbii 236	. . . . . . . 8 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
85	84	ex 418	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
86	85	vtocleg 3520	. . . . . 6 ⊢ (𝑌 ∈ 𝑋 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
87	86	com12 33	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
88	87	vtocleg 3520	. . . 4 ⊢ (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
89	88	pm2.43b 56	. . 3 ⊢ (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
90	89	pm2.43b 56	. 2 ⊢ (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
91	90	imp 412	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  Vcvv 3454  [wsbc 3742  ⦋csb 3851   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  ∪ ciun 4954   ↦ cmpt 5189  Oncon0 6369  Lim wlim 6370  suc csuc 6371  ‘cfv 6547  reccrdg 8431

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432

This theorem is used by: (None)