Theorem onfununi 8333

Description: A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
onfununi.1	⊢ (Lim 𝑦 → (𝐹‘𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐹‘𝑥))
onfununi.2	⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))

Assertion

Ref	Expression
onfununi	⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦   𝑥,𝑇

Allowed substitution hint:   𝑇(𝑦)

Proof of Theorem onfununi

Step	Hyp	Ref	Expression
1		ssorduni 7781	. . . . . . . . . 10 ⊢ (𝑆 ⊆ On → Ord ∪ 𝑆)
2	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) ∧ 𝑆 ≠ ∅) → Ord ∪ 𝑆)
3		nelneq 2886	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑆 ∧ ¬ ∪ 𝑆 ∈ 𝑆) → ¬ 𝑥 = ∪ 𝑆)
4		elssuni 4902	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ ∪ 𝑆)
5	4	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → 𝑥 ⊆ ∪ 𝑆)
6		ssel 3928	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑆 ⊆ On → (𝑥 ∈ 𝑆 → 𝑥 ∈ On))
7		eloni 6371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ On → Ord 𝑥)
8	6, 7	syl6 36	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆 ⊆ On → (𝑥 ∈ 𝑆 → Ord 𝑥))
9	8	imp 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → Ord 𝑥)
10		ordsseleq 6391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord 𝑥 ∧ Ord ∪ 𝑆) → (𝑥 ⊆ ∪ 𝑆 ↔ (𝑥 ∈ ∪ 𝑆 ∨ 𝑥 = ∪ 𝑆)))
11	9, 1, 10	syl2an 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ On) → (𝑥 ⊆ ∪ 𝑆 ↔ (𝑥 ∈ ∪ 𝑆 ∨ 𝑥 = ∪ 𝑆)))
12	11	anabss1 679	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → (𝑥 ⊆ ∪ 𝑆 ↔ (𝑥 ∈ ∪ 𝑆 ∨ 𝑥 = ∪ 𝑆)))
13	5, 12	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ ∪ 𝑆 ∨ 𝑥 = ∪ 𝑆))
14	13	ord 878	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → (¬ 𝑥 ∈ ∪ 𝑆 → 𝑥 = ∪ 𝑆))
15	14	con1d 146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → (¬ 𝑥 = ∪ 𝑆 → 𝑥 ∈ ∪ 𝑆))
16	3, 15	syl5 35	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ On ∧ 𝑥 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ∧ ¬ ∪ 𝑆 ∈ 𝑆) → 𝑥 ∈ ∪ 𝑆))
17	16	exp4b 436	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ On → (𝑥 ∈ 𝑆 → (𝑥 ∈ 𝑆 → (¬ ∪ 𝑆 ∈ 𝑆 → 𝑥 ∈ ∪ 𝑆))))
18	17	pm2.43d 54	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ On → (𝑥 ∈ 𝑆 → (¬ ∪ 𝑆 ∈ 𝑆 → 𝑥 ∈ ∪ 𝑆)))
19	18	com23 87	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ On → (¬ ∪ 𝑆 ∈ 𝑆 → (𝑥 ∈ 𝑆 → 𝑥 ∈ ∪ 𝑆)))
20	19	imp 412	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) → (𝑥 ∈ 𝑆 → 𝑥 ∈ ∪ 𝑆))
21	20	ssrdv 3940	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝑆)
22		ssn0 4358	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ∪ 𝑆 ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ≠ ∅)
23	21, 22	sylan 592	. . . . . . . . 9 ⊢ (((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ≠ ∅)
24	21	unissd 4880	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) → ∪ 𝑆 ⊆ ∪ ∪ 𝑆)
25		orduniss 6461	. . . . . . . . . . . . 13 ⊢ (Ord ∪ 𝑆 → ∪ ∪ 𝑆 ⊆ ∪ 𝑆)
26	1, 25	syl 18	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ On → ∪ ∪ 𝑆 ⊆ ∪ 𝑆)
27	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝑆)
28	24, 27	eqssd 3951	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) → ∪ 𝑆 = ∪ ∪ 𝑆)
29	28	adantr 486	. . . . . . . . 9 ⊢ (((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 = ∪ ∪ 𝑆)
30		df-lim 6366	. . . . . . . . 9 ⊢ (Lim ∪ 𝑆 ↔ (Ord ∪ 𝑆 ∧ ∪ 𝑆 ≠ ∅ ∧ ∪ 𝑆 = ∪ ∪ 𝑆))
31	2, 23, 29, 30	syl3anbrc 1362	. . . . . . . 8 ⊢ (((𝑆 ⊆ On ∧ ¬ ∪ 𝑆 ∈ 𝑆) ∧ 𝑆 ≠ ∅) → Lim ∪ 𝑆)
32	31	an32s 665	. . . . . . 7 ⊢ (((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → Lim ∪ 𝑆)
33	32	3adantl1 1185	. . . . . 6 ⊢ (((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → Lim ∪ 𝑆)
34		ssonuni 7782	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑇 → (𝑆 ⊆ On → ∪ 𝑆 ∈ On))
35		limeq 6373	. . . . . . . . . . . 12 ⊢ (𝑦 = ∪ 𝑆 → (Lim 𝑦 ↔ Lim ∪ 𝑆))
36		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∪ 𝑆 → (𝐹‘𝑦) = (𝐹‘∪ 𝑆))
37		iuneq1 4971	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∪ 𝑆 → ∪ 𝑥 ∈ 𝑦 (𝐹‘𝑥) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥))
38	36, 37	eqeq12d 2778	. . . . . . . . . . . 12 ⊢ (𝑦 = ∪ 𝑆 → ((𝐹‘𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐹‘𝑥) ↔ (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥)))
39	35, 38	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑦 = ∪ 𝑆 → ((Lim 𝑦 → (𝐹‘𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐹‘𝑥)) ↔ (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥))))
40		onfununi.1	. . . . . . . . . . 11 ⊢ (Lim 𝑦 → (𝐹‘𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐹‘𝑥))
41	39, 40	vtoclg 3520	. . . . . . . . . 10 ⊢ (∪ 𝑆 ∈ On → (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥)))
42	34, 41	syl6 36	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑇 → (𝑆 ⊆ On → (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥))))
43	42	imp 412	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On) → (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥)))
44	43	3adant3 1150	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥)))
45	44	adantr 486	. . . . . 6 ⊢ (((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → (Lim ∪ 𝑆 → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥)))
46	33, 45	mpd 16	. . . . 5 ⊢ (((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥))
47		eluni2 4874	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑆 ↔ ∃𝑦 ∈ 𝑆 𝑥 ∈ 𝑦)
48		ssel 3928	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ On → (𝑦 ∈ 𝑆 → 𝑦 ∈ On))
49	48	anim1d 623	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → (𝑦 ∈ On ∧ 𝑥 ∈ 𝑦)))
50		onelon 6386	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On)
51	49, 50	syl6 36	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On))
52	48	adantrd 497	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ On))
53		eloni 6371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On → Ord 𝑦)
54	48, 53	syl6 36	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ On → (𝑦 ∈ 𝑆 → Ord 𝑦))
55		ordelss 6377	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝑦 ∧ 𝑥 ∈ 𝑦) → 𝑥 ⊆ 𝑦)
56	55	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ On → ((Ord 𝑦 ∧ 𝑥 ∈ 𝑦) → 𝑥 ⊆ 𝑦))
57	54, 56	syland 615	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → 𝑥 ⊆ 𝑦))
58	51, 52, 57	3jcad 1147	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦)))
59		onfununi.2	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))
60	58, 59	syl6 36	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ On → ((𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
61	60	expd 421	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ On → (𝑦 ∈ 𝑆 → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
62	61	reximdvai 3175	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ On → (∃𝑦 ∈ 𝑆 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
63	47, 62	biimtrid 245	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ On → (𝑥 ∈ ∪ 𝑆 → ∃𝑦 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
64		ssiun 5009	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘𝑦) → (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦))
65	63, 64	syl6 36	. . . . . . . . . 10 ⊢ (𝑆 ⊆ On → (𝑥 ∈ ∪ 𝑆 → (𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦)))
66	65	ralrimiv 3155	. . . . . . . . 9 ⊢ (𝑆 ⊆ On → ∀𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦))
67		iunss 5007	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) ↔ ∀𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦))
68	66, 67	sylibr 237	. . . . . . . 8 ⊢ (𝑆 ⊆ On → ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦))
69		fveq2 6882	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
70	69	cbviunv 5001	. . . . . . . 8 ⊢ ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥)
71	68, 70	sseqtrdi 3974	. . . . . . 7 ⊢ (𝑆 ⊆ On → ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
72	71	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
73	72	adantr 486	. . . . 5 ⊢ (((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → ∪ 𝑥 ∈ ∪ 𝑆(𝐹‘𝑥) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
74	46, 73	eqsstrd 3968	. . . 4 ⊢ (((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ ∪ 𝑆 ∈ 𝑆) → (𝐹‘∪ 𝑆) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
75	74	ex 418	. . 3 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (¬ ∪ 𝑆 ∈ 𝑆 → (𝐹‘∪ 𝑆) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥)))
76		fveq2 6882	. . . 4 ⊢ (𝑥 = ∪ 𝑆 → (𝐹‘𝑥) = (𝐹‘∪ 𝑆))
77	76	ssiun2s 5011	. . 3 ⊢ (∪ 𝑆 ∈ 𝑆 → (𝐹‘∪ 𝑆) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
78	75, 77	pm2.61d2 183	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹‘∪ 𝑆) ⊆ ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))
79	34	imp 412	. . . . . 6 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On) → ∪ 𝑆 ∈ On)
80	79	3adant3 1150	. . . . 5 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ On)
81	6	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥 ∈ 𝑆 → 𝑥 ∈ On))
82	81, 4	jca2 523	. . . . 5 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥 ∈ 𝑆 → (𝑥 ∈ On ∧ 𝑥 ⊆ ∪ 𝑆)))
83		sseq2 3960	. . . . . . . 8 ⊢ (𝑦 = ∪ 𝑆 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ ∪ 𝑆))
84	83	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑆 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝑦) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ ∪ 𝑆)))
85	36	sseq2d 3966	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑆 → ((𝐹‘𝑥) ⊆ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆)))
86	84, 85	imbi12d 347	. . . . . 6 ⊢ (𝑦 = ∪ 𝑆 → (((𝑥 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑥 ⊆ ∪ 𝑆) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆))))
87	59	3com12 1141	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))
88	87	3expib 1140	. . . . . 6 ⊢ (𝑦 ∈ On → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
89	86, 88	vtoclga 3539	. . . . 5 ⊢ (∪ 𝑆 ∈ On → ((𝑥 ∈ On ∧ 𝑥 ⊆ ∪ 𝑆) → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆)))
90	80, 82, 89	sylsyld 62	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥 ∈ 𝑆 → (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆)))
91	90	ralrimiv 3155	. . 3 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆))
92		iunss 5007	. . 3 ⊢ (∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆))
93	91, 92	sylibr 237	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥) ⊆ (𝐹‘∪ 𝑆))
94	78, 93	eqssd 3951	1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐹‘𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ∪ ciun 4954  Ord word 6360  Oncon0 6361  Lim wlim 6362  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-iota 6493  df-fv 6545

This theorem is used by:  onovuni  8334