Theorem rdgssun 37292

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 3818	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
2		0ex 5328	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
3		rzal 4528	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
4		sbceq1a 3809	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → [∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
6	1, 2, 5	vtoclef 3571	. . . . . . . . . . 11 ⊢ [∅ / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)
7		vex 3486	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
8	7	elsuc 6464	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ suc 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
9		ssun1 4195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (rec(𝐹, 𝐴)‘𝑥) ⊆ ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
10		fvex 6932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (rec(𝐹, 𝐴)‘𝑥) ∈ V
11		rdgssun.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐵 ∈ V
12	11	csbex 5332	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵 ∈ V
13	10, 12	unex 7775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵) ∈ V
14		nfcv 2904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤𝐴
15		nfcv 2904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤𝑥
16		rdgssun.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))
17		nfmpt1 5277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑤(𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))
18	16, 17	nfcxfr 2902	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑤𝐹
19	18, 14	nfrdg 8466	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤rec(𝐹, 𝐴)
20	19, 15	nffv 6929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤(rec(𝐹, 𝐴)‘𝑥)
21	20	nfcsb1 3939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵
22	20, 21	nfun 4187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
23		rdgeq1 8463	. . . . . . . . . . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝑤 = (rec(𝐹, 𝐴)‘𝑥))
26		csbeq1a 3929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (rec(𝐹, 𝐴)‘𝑥) → 𝐵 = ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵)
27	25, 26	uneq12d 4186	. . . . . . . . . . . . . . . . . . . . . 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 8480	. . . . . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑥) = ((rec(𝐹, 𝐴)‘𝑥) ∪ ⦋(rec(𝐹, 𝐴)‘𝑥) / 𝑤⦌𝐵))
30	9, 29	sseqtrrid 4056	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → (rec(𝐹, 𝐴)‘𝑥) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥))
31		sstr2 4009	. . . . . . . . . . . . . . . . . . 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 31	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
33	32	imim2d 57	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑥))
36	35	sseq1d 4034	. . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑦 = 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
38	37	adantr 480	. . . . . . . . . . . . . . . 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 858	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 3165	. . . . . . . . . . . 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 3799	. . . . . . . . . . . . 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 3486	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
45	44	sucex 7838	. . . . . . . . . . . . . 14 ⊢ suc 𝑥 ∈ V
46		fveq2 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘suc 𝑥))
47	46	sseq2d 4035	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘suc 𝑥)))
48	47	raleqbi1dv 3341	. . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑧))
50	49	sseq2d 4035	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧)))
51	50	raleqbi1dv 3341	. . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . 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 3693	. . . . . . . . . . . . 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 275	. . . . . . . . . . . 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 252	. . . . . . . . . . 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 5073	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑧 → (rec(𝐹, 𝐴)‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
57	56	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
58		vex 3486	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
59		rdglim2a 8485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
60	58, 59	mpan 689	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑧 → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
61	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑧) = ∪ 𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦))
62	57, 61	sseqtrrd 4044	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝑧 ∧ 𝑦 ∈ 𝑧) → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
63	62	ralrimiva 3148	. . . . . . . . . . . . 13 ⊢ (Lim 𝑧 → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑧))
64		df-sbc 3799	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . 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 2715	. . . . . . . . . . . . . 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 275	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
70	69	a1d 25	. . . . . . . . . . 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 7896	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥))
72		rsp 3248	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (rec(𝐹, 𝐴)‘𝑦) ⊆ (rec(𝐹, 𝐴)‘𝑥)))
74		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ On ↔ 𝑋 ∈ On))
75	74	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑥 ∈ On ↔ 𝑋 ∈ On))
76		eleq12 2828	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝑥 ↔ 𝑌 ∈ 𝑋))
77		fveq2 6919	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
78	77	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑌))
79		fveq2 6919	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
80	79	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝑋))
81	78, 80	sseq12d 4036	. . . . . . . . . . 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 344	. . . . . . . . . 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 344	. . . . . . . . 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 233	. . . . . . . 8 ⊢ ((𝑦 = 𝑌 ∧ 𝑥 = 𝑋) → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
85	84	ex 412	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
86	85	vtocleg 3560	. . . . . 6 ⊢ (𝑌 ∈ 𝑋 → (𝑥 = 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
87	86	com12 32	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
88	87	vtocleg 3560	. . . 4 ⊢ (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))))
89	88	pm2.43b 55	. . 3 ⊢ (𝑌 ∈ 𝑋 → (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))))
90	89	pm2.43b 55	. 2 ⊢ (𝑋 ∈ On → (𝑌 ∈ 𝑋 → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)))
91	90	imp 406	1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2103  {cab 2711  ∀wral 3063  Vcvv 3482  [wsbc 3798  ⦋csb 3915   ∪ cun 3968   ⊆ wss 3970  ∅c0 4347  ∪ ciun 5019   ↦ cmpt 5252  Oncon0 6394  Lim wlim 6395  suc csuc 6396  ‘cfv 6572  reccrdg 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462

This theorem is referenced by: (None)