Theorem ingru 9637

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 3807	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ 𝐴) = (𝑈 ∩ 𝐴))
2	1	eleq1d 2686	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ 𝐴) ∈ Univ ↔ (𝑈 ∩ 𝐴) ∈ Univ))
3	2	imbi2d 330	. . 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 9614	. . . . . 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 256	. . . . 5 ⊢ (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑢)))
6		trin 4763	. . . . . . 7 ⊢ ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢 ∩ 𝐴))
7	6	ex 450	. . . . . 6 ⊢ (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢 ∩ 𝐴)))
8		inss1 3833	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝑢
9		ssralv 3666	. . . . . . . 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 3834	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝐴
12		ssralv 3666	. . . . . . . 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 3796	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑥 ∈ (𝑢 ∩ 𝐴) ↔ (𝒫 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝐴))
15	14	simplbi2 655	. . . . . . . . . . . 12 ⊢ (𝒫 𝑥 ∈ 𝑢 → (𝒫 𝑥 ∈ 𝐴 → 𝒫 𝑥 ∈ (𝑢 ∩ 𝐴)))
16		ssralv 3666	. . . . . . . . . . . . . 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 3666	. . . . . . . . . . . . . 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 3796	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
21	20	simplbi2 655	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢 ∩ 𝐴)))
22	21	ral2imi 2947	. . . . . . . . . . . . 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 880	. . . . . . . . . . 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 3203	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
26		mapss 7900	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ V ∧ (𝑢 ∩ 𝐴) ⊆ 𝑢) → ((𝑢 ∩ 𝐴) ↑𝑚 𝑥) ⊆ (𝑢 ↑𝑚 𝑥))
27	25, 8, 26	mp2an 708	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥) ⊆ (𝑢 ↑𝑚 𝑥)
28		ssralv 3666	. . . . . . . . . . . . 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 4799	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝐴) ∈ V
31		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	30, 31	elmap 7886	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥) ↔ 𝑦:𝑥⟶(𝑢 ∩ 𝐴))
33		fss 6056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦:𝑥⟶(𝑢 ∩ 𝐴) ∧ (𝑢 ∩ 𝐴) ⊆ 𝐴) → 𝑦:𝑥⟶𝐴)
34	11, 33	mpan2 707	. . . . . . . . . . . . . . . 16 ⊢ (𝑦:𝑥⟶(𝑢 ∩ 𝐴) → 𝑦:𝑥⟶𝐴)
35	32, 34	sylbi 207	. . . . . . . . . . . . . . 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 1739	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦(𝑦 ∈ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
38		df-ral 2917	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥) → ∪ ran 𝑦 ∈ 𝐴))
39	37, 38	sylibr 224	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴) → ∀𝑦 ∈ ((𝑢 ∩ 𝐴) ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝐴)
40		elin 3796	. . . . . . . . . . . . . 14 ⊢ (∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴) ↔ (∪ ran 𝑦 ∈ 𝑢 ∧ ∪ ran 𝑦 ∈ 𝐴))
41	40	simplbi2 655	. . . . . . . . . . . . 13 ⊢ (∪ ran 𝑦 ∈ 𝑢 → (∪ ran 𝑦 ∈ 𝐴 → ∪ ran 𝑦 ∈ (𝑢 ∩ 𝐴)))
42	41	ral2imi 2947	. . . . . . . . . . . 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 880	. . . . . . . . . 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 1259	. . . . . . . . 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 1039	. . . . . . . . 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 1039	. . . . . . . . 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 285	. . . . . . . 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 2947	. . . . . . 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 880	. . . . 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 9614	. . . . 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 242	. . 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 3272	. 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 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {cpr 4179  ∪ cuni 4436  Tr wtr 4752  ran crn 5115  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857  Univcgru 9612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-gru 9613

This theorem is referenced by:  wfgru  9638