Theorem ingru 10795

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 4166	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ 𝐴) = (𝑈 ∩ 𝐴))
2	1	eleq1d 2848	. . . 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 10772	. . . . . 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 270	. . . . 5 ⊢ (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)))
6		trin 5230	. . . . . . 7 ⊢ ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢 ∩ 𝐴))
7	6	ex 417	. . . . . 6 ⊢ (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢 ∩ 𝐴)))
8		inss1 4189	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝑢
9		ssralv 4006	. . . . . . . 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 4190	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝐴
12		ssralv 4006	. . . . . . . 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 3921	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ↔ (𝒫 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝐴))
15	14	simplbi2 505	. . . . . . . . . . . 12 ⊢ (𝒫 𝑥 ∈ 𝑢 → (𝒫 𝑥 ∈ 𝐴 → 𝒫 𝑥 ∈ (𝑢 ∩ 𝐴)))
16		ssralv 4006	. . . . . . . . . . . . . 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 4006	. . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
21	20	simplbi2 505	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
22	21	ral2imi 3104	. . . . . . . . . . . . 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 41	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 → (∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢 ∩ 𝐴){𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
24	15, 23	im2anan9 631	. . . . . . . . . . 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 3459	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
26		mapss 8883	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ V ∧ (𝑢 ∩ 𝐴) ⊆ 𝑢) → ((𝑢 ∩ 𝐴) ↑m 𝑥) ⊆ (𝑢 ↑m 𝑥))
27	25, 8, 26	mp2an 704	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝐴) ↑m 𝑥) ⊆ (𝑢 ↑m 𝑥)
28		ssralv 4006	. . . . . . . . . . . . 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 5286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝐴) ∈ V
31		vex 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	30, 31	elmap 8865	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) ↔ 𝑦:𝑥⟶(𝑢 ∩ 𝐴))
33		fss 6722	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦:𝑥⟶(𝑢 ∩ 𝐴) ∧ (𝑢 ∩ 𝐴) ⊆ 𝐴) → 𝑦:𝑥⟶𝐴)
34	11, 33	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑦:𝑥⟶(𝑢 ∩ 𝐴) → 𝑦:𝑥⟶𝐴)
35	32, 34	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → 𝑦:𝑥⟶𝐴)
36	35	imim1i 64	. . . . . . . . . . . . . 14 ⊢ ((𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
37	36	alimi 1841	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦(𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
38	37	ralrid 3087	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ 𝐴)
39		elin 3921	. . . . . . . . . . . . . 14 ⊢ (∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴) ↔ (∪ ran 𝑦 ∈ 𝑢 ∧ ∪ ran 𝑦 ∈ 𝐴))
40	39	simplbi2 505	. . . . . . . . . . . . 13 ⊢ (∪ ran 𝑦 ∈ 𝑢 → (∪ ran 𝑦 ∈ 𝐴 → ∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
41	40	ral2imi 3104	. . . . . . . . . . . 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)))
42	29, 38, 41	syl2im 41	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 → (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑m 𝑥)∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
43	24, 42	im2anan9 631	. . . . . . . . . 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))))
44	43	3impa 1127	. . . . . . . . 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))))
45		df-3an 1105	. . . . . . . . 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)))
46		df-3an 1105	. . . . . . . . 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)))
47	44, 45, 46	3imtr4g 299	. . . . . . . 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))))
48	47	ral2imi 3104	. . . . . . 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))))
49	10, 13, 48	syl2im 41	. . . . . 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))))
50	7, 49	im2anan9 631	. . . . 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)))))
51	5, 50	syl 18	. . . 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)))))
52		elgrug 10772	. . . . 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)))))
53	30, 52	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))))
54	51, 53	imbitrrdi 255	. . 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))
55	3, 54	vtoclga 3541	. 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))
56	55	com12 33	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 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  {cpr 4591  ∪ cuni 4872  Tr wtr 5218  ran crn 5662  ⟶wf 6532  (class class class)co 7410   ↑_m cmap 8820  Univcgru 10770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-gru 10771

This theorem is referenced by:  wfgru  10796