Theorem fiiuncl 44053

Description: If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
fiiuncl.xph	⊢ Ⅎ𝑥𝜑
fiiuncl.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)
fiiuncl.un	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷)
fiiuncl.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fiiuncl.n0	⊢ (𝜑 → 𝐴 ≠ ∅)

Assertion

Ref	Expression
fiiuncl	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵,𝑧   𝑥,𝐷,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑦,𝑧)   𝐵(𝑥)

Proof of Theorem fiiuncl

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fiiuncl.n0	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		neeq1 3001	. . . 4 ⊢ (𝑣 = ∅ → (𝑣 ≠ ∅ ↔ ∅ ≠ ∅))
3		iuneq1 5012	. . . . 5 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
4	3	eleq1d 2816	. . . 4 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷 ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ 𝐷))
5	2, 4	imbi12d 343	. . 3 ⊢ (𝑣 = ∅ → ((𝑣 ≠ ∅ → ∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷) ↔ (∅ ≠ ∅ → ∪ 𝑥 ∈ ∅ 𝐵 ∈ 𝐷)))
6		neeq1 3001	. . . 4 ⊢ (𝑣 = 𝑤 → (𝑣 ≠ ∅ ↔ 𝑤 ≠ ∅))
7		iuneq1 5012	. . . . 5 ⊢ (𝑣 = 𝑤 → ∪ 𝑥 ∈ 𝑣 𝐵 = ∪ 𝑥 ∈ 𝑤 𝐵)
8	7	eleq1d 2816	. . . 4 ⊢ (𝑣 = 𝑤 → (∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷 ↔ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷))
9	6, 8	imbi12d 343	. . 3 ⊢ (𝑣 = 𝑤 → ((𝑣 ≠ ∅ → ∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷) ↔ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)))
10		neeq1 3001	. . . 4 ⊢ (𝑣 = (𝑤 ∪ {𝑢}) → (𝑣 ≠ ∅ ↔ (𝑤 ∪ {𝑢}) ≠ ∅))
11		iuneq1 5012	. . . . 5 ⊢ (𝑣 = (𝑤 ∪ {𝑢}) → ∪ 𝑥 ∈ 𝑣 𝐵 = ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵)
12	11	eleq1d 2816	. . . 4 ⊢ (𝑣 = (𝑤 ∪ {𝑢}) → (∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷 ↔ ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷))
13	10, 12	imbi12d 343	. . 3 ⊢ (𝑣 = (𝑤 ∪ {𝑢}) → ((𝑣 ≠ ∅ → ∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷) ↔ ((𝑤 ∪ {𝑢}) ≠ ∅ → ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷)))
14		neeq1 3001	. . . 4 ⊢ (𝑣 = 𝐴 → (𝑣 ≠ ∅ ↔ 𝐴 ≠ ∅))
15		iuneq1 5012	. . . . 5 ⊢ (𝑣 = 𝐴 → ∪ 𝑥 ∈ 𝑣 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
16	15	eleq1d 2816	. . . 4 ⊢ (𝑣 = 𝐴 → (∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷))
17	14, 16	imbi12d 343	. . 3 ⊢ (𝑣 = 𝐴 → ((𝑣 ≠ ∅ → ∪ 𝑥 ∈ 𝑣 𝐵 ∈ 𝐷) ↔ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷)))
18		neirr 2947	. . . . 5 ⊢ ¬ ∅ ≠ ∅
19	18	pm2.21i 119	. . . 4 ⊢ (∅ ≠ ∅ → ∪ 𝑥 ∈ ∅ 𝐵 ∈ 𝐷)
20	19	a1i 11	. . 3 ⊢ (𝜑 → (∅ ≠ ∅ → ∪ 𝑥 ∈ ∅ 𝐵 ∈ 𝐷))
21		iunxun 5096	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 = (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ∪ 𝑥 ∈ {𝑢}𝐵)
22		nfcsb1v 3917	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
23		vex 3476	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
24		csbeq1a 3906	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
25	22, 23, 24	iunxsnf 44052	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑢}𝐵 = ⦋𝑢 / 𝑥⦌𝐵
26	25	uneq2i 4159	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ∪ 𝑥 ∈ {𝑢}𝐵) = (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵)
27	21, 26	eqtri 2758	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 = (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵)
28		iuneq1 5012	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
29		0iun 5065	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ ∅ 𝐵 = ∅)
31	28, 30	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∅)
32	31	uneq1d 4161	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) = (∅ ∪ ⦋𝑢 / 𝑥⦌𝐵))
33		0un 4391	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ ⦋𝑢 / 𝑥⦌𝐵) = ⦋𝑢 / 𝑥⦌𝐵
34		unidm 4151	. . . . . . . . . . . . . 14 ⊢ (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) = ⦋𝑢 / 𝑥⦌𝐵
35	33, 34	eqtr4i 2761	. . . . . . . . . . . . 13 ⊢ (∅ ∪ ⦋𝑢 / 𝑥⦌𝐵) = (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵)
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (∅ ∪ ⦋𝑢 / 𝑥⦌𝐵) = (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵))
37	32, 36	eqtrd 2770	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) = (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵))
38	37	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ 𝑤 = ∅) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) = (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵))
39		simpl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) → 𝜑)
40		eldifi 4125	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝐴 ∖ 𝑤) → 𝑢 ∈ 𝐴)
41	40	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) → 𝑢 ∈ 𝐴)
42		fiiuncl.xph	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝜑
43		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑢 ∈ 𝐴
44	42, 43	nfan 1900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑢 ∈ 𝐴)
45		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐷
46	22, 45	nfel 2915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷
47	44, 46	nfim 1897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑢 ∈ 𝐴) → ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷)
48		eleq1 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
49	48	anbi2d 627	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑢 ∈ 𝐴)))
50	24	eleq1d 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝐵 ∈ 𝐷 ↔ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷))
51	49, 50	imbi12d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷) ↔ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷)))
52		fiiuncl.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)
53	47, 51, 52	chvarfv 2231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷)
54	34, 53	eqeltrid 2835	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
55	39, 41, 54	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) → (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
56	55	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ 𝑤 = ∅) → (⦋𝑢 / 𝑥⦌𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
57	38, 56	eqeltrd 2831	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ 𝑤 = ∅) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
58	57	adantlr 711	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) ∧ 𝑤 = ∅) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
59		simplll 771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) ∧ ¬ 𝑤 = ∅) → 𝜑)
60	40	ad3antlr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) ∧ ¬ 𝑤 = ∅) → 𝑢 ∈ 𝐴)
61		neqne 2946	. . . . . . . . . . . 12 ⊢ (¬ 𝑤 = ∅ → 𝑤 ≠ ∅)
62	61	adantl 480	. . . . . . . . . . 11 ⊢ (((𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) ∧ ¬ 𝑤 = ∅) → 𝑤 ≠ ∅)
63		simpl 481	. . . . . . . . . . 11 ⊢ (((𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) ∧ ¬ 𝑤 = ∅) → (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷))
64	62, 63	mpd 15	. . . . . . . . . 10 ⊢ (((𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) ∧ ¬ 𝑤 = ∅) → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)
65	64	adantll 710	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) ∧ ¬ 𝑤 = ∅) → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)
66	53	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷)
67		simp3 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)
68		simp1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → 𝜑)
69	68, 67, 66	3jca 1126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → (𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷))
70		eleq1 2819	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → (𝑧 ∈ 𝐷 ↔ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷))
71	70	3anbi3d 1440	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) ↔ (𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷)))
72		uneq2 4156	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) = (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵))
73	72	eleq1d 2816	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → ((∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷 ↔ (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷))
74	71, 73	imbi12d 343	. . . . . . . . . . . 12 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → (((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷) ↔ ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)))
75	74	imbi2d 339	. . . . . . . . . . 11 ⊢ (𝑧 = ⦋𝑢 / 𝑥⦌𝐵 → ((∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 → ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷)) ↔ (∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 → ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷))))
76		eleq1 2819	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝑤 𝐵 → (𝑦 ∈ 𝐷 ↔ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷))
77	76	3anbi2d 1439	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝑤 𝐵 → ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) ↔ (𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷)))
78		uneq1 4155	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝑤 𝐵 → (𝑦 ∪ 𝑧) = (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧))
79	78	eleq1d 2816	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝑤 𝐵 → ((𝑦 ∪ 𝑧) ∈ 𝐷 ↔ (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷))
80	77, 79	imbi12d 343	. . . . . . . . . . . 12 ⊢ (𝑦 = ∪ 𝑥 ∈ 𝑤 𝐵 → (((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷) ↔ ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷)))
81		fiiuncl.un	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷)
82	80, 81	vtoclg 3541	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 → ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ 𝑧) ∈ 𝐷))
83	75, 82	vtoclg 3541	. . . . . . . . . 10 ⊢ (⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 → ((𝜑 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷 ∧ ⦋𝑢 / 𝑥⦌𝐵 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)))
84	66, 67, 69, 83	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
85	59, 60, 65, 84	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) ∧ ¬ 𝑤 = ∅) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
86	58, 85	pm2.61dan 809	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) → (∪ 𝑥 ∈ 𝑤 𝐵 ∪ ⦋𝑢 / 𝑥⦌𝐵) ∈ 𝐷)
87	27, 86	eqeltrid 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) → ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷)
88	87	a1d 25	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) ∧ (𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷)) → ((𝑤 ∪ {𝑢}) ≠ ∅ → ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷))
89	88	ex 411	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤)) → ((𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → ((𝑤 ∪ {𝑢}) ≠ ∅ → ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷)))
90	89	adantrl 712	. . 3 ⊢ ((𝜑 ∧ (𝑤 ⊆ 𝐴 ∧ 𝑢 ∈ (𝐴 ∖ 𝑤))) → ((𝑤 ≠ ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ 𝐷) → ((𝑤 ∪ {𝑢}) ≠ ∅ → ∪ 𝑥 ∈ (𝑤 ∪ {𝑢})𝐵 ∈ 𝐷)))
91		fiiuncl.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
92	5, 9, 13, 17, 20, 90, 91	findcard2d 9168	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷))
93	1, 92	mpd 15	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104   ≠ wne 2938  ⦋csb 3892   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  {csn 4627  ∪ ciun 4996  Fincfn 8941

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-en 8942  df-fin 8945

This theorem is referenced by:  fiunicl  44055  caragenfiiuncl  45529