Theorem ingru 10240

Description: The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
ingru	⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem ingru

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 4184	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ 𝐴) = (𝑈 ∩ 𝐴))
2	1	eleq1d 2900	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ 𝐴) ∈ Univ ↔ (𝑈 ∩ 𝐴) ∈ Univ))
3	2	imbi2d 343	. . 3 ⊢ (𝑢 = 𝑈 → (((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑢 ∩ 𝐴) ∈ Univ) ↔ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∩ 𝐴) ∈ Univ)))
4		elgrug 10217	. . . . . 6 ⊢ (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))))
5	4	ibi 269	. . . . 5 ⊢ (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)))
6		trin 5185	. . . . . . 7 ⊢ ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢 ∩ 𝐴))
7	6	ex 415	. . . . . 6 ⊢ (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢 ∩ 𝐴)))
8		inss1 4208	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝑢
9		ssralv 4036	. . . . . . . 8 ⊢ ((𝑢 ∩ 𝐴) ⊆ 𝑢 → (∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))
11		inss2 4209	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝐴
12		ssralv 4036	. . . . . . . 8 ⊢ ((𝑢 ∩ 𝐴) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))))
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)))
14		elin 4172	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ↔ (𝒫 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝐴))
15	14	simplbi2 503	. . . . . . . . . . . 12 ⊢ (𝒫 𝑥 ∈ 𝑢 → (𝒫 𝑥 ∈ 𝐴 → 𝒫 𝑥 ∈ (𝑢 ∩ 𝐴)))
16		ssralv 4036	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝐴) ⊆ 𝑢 → (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝑢))
17	8, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝑢)
18		ssralv 4036	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝐴) ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝐴))
19	11, 18	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝐴)
20		elin 4172	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
21	20	simplbi2 503	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
22	21	ral2imi 3159	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝑢 → (∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
23	17, 19, 22	syl2im 40	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 → (∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
24	15, 23	im2anan9 621	. . . . . . . . . . 11 ⊢ ((𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴))))
25		vex 3500	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
26		mapss 8456	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ V ∧ (𝑢 ∩ 𝐴) ⊆ 𝑢) → ((𝑢 ∩ 𝐴) ↑m 𝑥) ⊆ (𝑢 ↑m 𝑥))
27	25, 8, 26	mp2an 690	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝐴) ↑m 𝑥) ⊆ (𝑢 ↑m 𝑥)
28		ssralv 4036	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∩ 𝐴) ↑m 𝑥) ⊆ (𝑢 ↑m 𝑥) → (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)
30	25	inex1 5224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝐴) ∈ V
31		vex 3500	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	30, 31	elmap 8438	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) ↔ 𝑦:𝑥⟶(𝑢 ∩ 𝐴))
33		fss 6530	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦:𝑥⟶(𝑢 ∩ 𝐴) ∧ (𝑢 ∩ 𝐴) ⊆ 𝐴) → 𝑦:𝑥⟶𝐴)
34	11, 33	mpan2 689	. . . . . . . . . . . . . . . 16 ⊢ (𝑦:𝑥⟶(𝑢 ∩ 𝐴) → 𝑦:𝑥⟶𝐴)
35	32, 34	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → 𝑦:𝑥⟶𝐴)
36	35	imim1i 63	. . . . . . . . . . . . . 14 ⊢ ((𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
37	36	alimi 1811	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦(𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
38		df-ral 3146	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
39	37, 38	sylibr 236	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝐴)
40		elin 4172	. . . . . . . . . . . . . 14 ⊢ (∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴) ↔ (∪ ran 𝑦 ∈ 𝑢 ∧ ∪ ran 𝑦 ∈ 𝐴))
41	40	simplbi2 503	. . . . . . . . . . . . 13 ⊢ (∪ ran 𝑦 ∈ 𝑢 → (∪ ran 𝑦 ∈ 𝐴 → ∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
42	41	ral2imi 3159	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 → (∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝐴 → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
43	29, 39, 42	syl2im 40	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 → (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
44	24, 43	im2anan9 621	. . . . . . . . . 10 ⊢ (((𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢) ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → (((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ((𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
45	44	3impa 1106	. . . . . . . . 9 ⊢ ((𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → (((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ((𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
46		df-3an 1085	. . . . . . . . 9 ⊢ ((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) ↔ ((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)))
47		df-3an 1085	. . . . . . . . 9 ⊢ ((𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)) ↔ ((𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
48	45, 46, 47	3imtr4g 298	. . . . . . . 8 ⊢ ((𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → ((𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → (𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
49	48	ral2imi 3159	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → (∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
50	10, 13, 49	syl2im 40	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) → (∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴)) → ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
51	7, 50	im2anan9 621	. . . . 5 ⊢ ((Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)) → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (Tr (𝑢 ∩ 𝐴) ∧ ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))))
52	5, 51	syl 17	. . . 4 ⊢ (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (Tr (𝑢 ∩ 𝐴) ∧ ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))))
53		elgrug 10217	. . . . 5 ⊢ ((𝑢 ∩ 𝐴) ∈ V → ((𝑢 ∩ 𝐴) ∈ Univ ↔ (Tr (𝑢 ∩ 𝐴) ∧ ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))))
54	30, 53	ax-mp 5	. . . 4 ⊢ ((𝑢 ∩ 𝐴) ∈ Univ ↔ (Tr (𝑢 ∩ 𝐴) ∧ ∀𝑥 ∈ (𝑢 ∩ 𝐴)(𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ∧ ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴))))
55	52, 54	syl6ibr 254	. . 3 ⊢ (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑢 ∩ 𝐴) ∈ Univ))
56	3, 55	vtoclga 3577	. 2 ⊢ (𝑈 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∩ 𝐴) ∈ Univ))
57	56	com12 32	1 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∀wral 3141  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  {cpr 4572  ∪ cuni 4841  Tr wtr 5175  ran crn 5559  ⟶wf 6354  (class class class)co 7159   ↑_m cmap 8409  Univcgru 10215

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411  df-gru 10216

This theorem is referenced by:  wfgru  10241