Theorem trust 22033

Description: The trace of a uniform structure 𝑈 on a subset 𝐴 is a uniform structure on 𝐴. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)

Assertion

Ref	Expression
trust	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))

Proof of Theorem trust

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

Step	Hyp	Ref	Expression
1		restsspw 16092	. . . 4 ⊢ (𝑈 ↾t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴)
2	1	a1i 11	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴))
3		inxp 5254	. . . . . 6 ⊢ ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = ((𝑋 ∩ 𝐴) × (𝑋 ∩ 𝐴))
4		sseqin2 3817	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
5	4	biimpi 206	. . . . . . 7 ⊢ (𝐴 ⊆ 𝑋 → (𝑋 ∩ 𝐴) = 𝐴)
6	5	sqxpeqd 5141	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 → ((𝑋 ∩ 𝐴) × (𝑋 ∩ 𝐴)) = (𝐴 × 𝐴))
7	3, 6	syl5eq 2668	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
8	7	adantl 482	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9		simpl 473	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
10		elfvex 6221	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V)
12		simpr 477	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
13	11, 12	ssexd 4805	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
14		xpexg 6960	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
15	13, 13, 14	syl2anc 693	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ V)
16		ustbasel 22010	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
17	16	adantr 481	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
18		elrestr 16089	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ (𝑋 × 𝑋) ∈ 𝑈) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
19	9, 15, 17, 18	syl3anc 1326	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
20	8, 19	eqeltrrd 2702	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
21	9	ad5antr 770	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
22	15	ad5antr 770	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
23		simplr 792	. . . . . . . . . . 11 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝑈)
24		simp-4r 807	. . . . . . . . . . . . . 14 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ 𝒫 (𝐴 × 𝐴))
25	24	elpwid 4170	. . . . . . . . . . . . 13 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ⊆ (𝐴 × 𝐴))
26	12	ad5antr 770	. . . . . . . . . . . . . 14 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋)
27		xpss12 5225	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
28	26, 26, 27	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
29	25, 28	sstrd 3613	. . . . . . . . . . . 12 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ⊆ (𝑋 × 𝑋))
30		ustssxp 22008	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
31	21, 23, 30	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢 ⊆ (𝑋 × 𝑋))
32	29, 31	unssd 3789	. . . . . . . . . . 11 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∪ 𝑢) ⊆ (𝑋 × 𝑋))
33		ssun2 3777	. . . . . . . . . . . 12 ⊢ 𝑢 ⊆ (𝑤 ∪ 𝑢)
34		ustssel 22009	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈 ∧ (𝑤 ∪ 𝑢) ⊆ (𝑋 × 𝑋)) → (𝑢 ⊆ (𝑤 ∪ 𝑢) → (𝑤 ∪ 𝑢) ∈ 𝑈))
35	33, 34	mpi 20	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈 ∧ (𝑤 ∪ 𝑢) ⊆ (𝑋 × 𝑋)) → (𝑤 ∪ 𝑢) ∈ 𝑈)
36	21, 23, 32, 35	syl3anc 1326	. . . . . . . . . 10 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∪ 𝑢) ∈ 𝑈)
37		df-ss 3588	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ (𝐴 × 𝐴) ↔ (𝑤 ∩ (𝐴 × 𝐴)) = 𝑤)
38	25, 37	sylib 208	. . . . . . . . . . . . 13 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∩ (𝐴 × 𝐴)) = 𝑤)
39	38	uneq1d 3766	. . . . . . . . . . . 12 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴))) = (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))))
40		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
41		simpllr 799	. . . . . . . . . . . . . 14 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 ⊆ 𝑤)
42	40, 41	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑤)
43		ssequn2 3786	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑤 ↔ (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))) = 𝑤)
44	42, 43	sylib 208	. . . . . . . . . . . 12 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))) = 𝑤)
45	39, 44	eqtr2d 2657	. . . . . . . . . . 11 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 = ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴))))
46		indir 3875	. . . . . . . . . . 11 ⊢ ((𝑤 ∪ 𝑢) ∩ (𝐴 × 𝐴)) = ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴)))
47	45, 46	syl6eqr 2674	. . . . . . . . . 10 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 = ((𝑤 ∪ 𝑢) ∩ (𝐴 × 𝐴)))
48		ineq1 3807	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 ∪ 𝑢) → (𝑥 ∩ (𝐴 × 𝐴)) = ((𝑤 ∪ 𝑢) ∩ (𝐴 × 𝐴)))
49	48	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ∪ 𝑢) → (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) ↔ 𝑤 = ((𝑤 ∪ 𝑢) ∩ (𝐴 × 𝐴))))
50	49	rspcev 3309	. . . . . . . . . 10 ⊢ (((𝑤 ∪ 𝑢) ∈ 𝑈 ∧ 𝑤 = ((𝑤 ∪ 𝑢) ∩ (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
51	36, 47, 50	syl2anc 693	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
52		elrest 16088	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
53	52	biimpar 502	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
54	21, 22, 51, 53	syl21anc 1325	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
55		elrest 16088	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))))
56	55	biimpa 501	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
57	15, 56	syldanl 735	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
58	57	ad2antrr 762	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) → ∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
59	54, 58	r19.29a 3078	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
60	59	ex 450	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))))
61	60	ralrimiva 2966	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))))
62	9	ad5antr 770	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
63	15	ad5antr 770	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝐴 × 𝐴) ∈ V)
64		simpllr 799	. . . . . . . . . 10 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑢 ∈ 𝑈)
65		simplr 792	. . . . . . . . . 10 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑥 ∈ 𝑈)
66		ustincl 22011	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (𝑢 ∩ 𝑥) ∈ 𝑈)
67	62, 64, 65, 66	syl3anc 1326	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑢 ∩ 𝑥) ∈ 𝑈)
68		simprl 794	. . . . . . . . . . 11 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
69		simprr 796	. . . . . . . . . . 11 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
70	68, 69	ineq12d 3815	. . . . . . . . . 10 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣 ∩ 𝑤) = ((𝑢 ∩ (𝐴 × 𝐴)) ∩ (𝑥 ∩ (𝐴 × 𝐴))))
71		inindir 3831	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝑥) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∩ (𝑥 ∩ (𝐴 × 𝐴)))
72	70, 71	syl6eqr 2674	. . . . . . . . 9 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣 ∩ 𝑤) = ((𝑢 ∩ 𝑥) ∩ (𝐴 × 𝐴)))
73		ineq1 3807	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ 𝑥) → (𝑦 ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ 𝑥) ∩ (𝐴 × 𝐴)))
74	73	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑦 = (𝑢 ∩ 𝑥) → ((𝑣 ∩ 𝑤) = (𝑦 ∩ (𝐴 × 𝐴)) ↔ (𝑣 ∩ 𝑤) = ((𝑢 ∩ 𝑥) ∩ (𝐴 × 𝐴))))
75	74	rspcev 3309	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑥) ∈ 𝑈 ∧ (𝑣 ∩ 𝑤) = ((𝑢 ∩ 𝑥) ∩ (𝐴 × 𝐴))) → ∃𝑦 ∈ 𝑈 (𝑣 ∩ 𝑤) = (𝑦 ∩ (𝐴 × 𝐴)))
76	67, 72, 75	syl2anc 693	. . . . . . . 8 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → ∃𝑦 ∈ 𝑈 (𝑣 ∩ 𝑤) = (𝑦 ∩ (𝐴 × 𝐴)))
77		elrest 16088	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → ((𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑦 ∈ 𝑈 (𝑣 ∩ 𝑤) = (𝑦 ∩ (𝐴 × 𝐴))))
78	77	biimpar 502	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑦 ∈ 𝑈 (𝑣 ∩ 𝑤) = (𝑦 ∩ (𝐴 × 𝐴))) → (𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
79	62, 63, 76, 78	syl21anc 1325	. . . . . . 7 ⊢ (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
80	57	adantr 481	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
81	9	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
82	15	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
83		simpr 477	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
84	52	biimpa 501	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
85	81, 82, 83, 84	syl21anc 1325	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
86		reeanv 3107	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑈 ∃𝑥 ∈ 𝑈 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))) ↔ (∃𝑢 ∈ 𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ ∃𝑥 ∈ 𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
87	80, 85, 86	sylanbrc 698	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑢 ∈ 𝑈 ∃𝑥 ∈ 𝑈 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
88	79, 87	r19.29vva 3081	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
89	88	ralrimiva 2966	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
90		simp-4l 806	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
91		simplr 792	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝑈)
92		ustdiag 22012	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑢)
93	90, 91, 92	syl2anc 693	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝑋) ⊆ 𝑢)
94		simp-4r 807	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋)
95		inss1 3833	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ ( I ↾ 𝑋)
96		resss 5422	. . . . . . . . . . . . . 14 ⊢ ( I ↾ 𝑋) ⊆ I
97	95, 96	sstri 3612	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ I
98		iss 5447	. . . . . . . . . . . . 13 ⊢ ((( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ I ↔ (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴))))
99	97, 98	mpbi 220	. . . . . . . . . . . 12 ⊢ (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
100		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
101		ssel2 3598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝑋)
102		equid 1939	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 = 𝑢
103		resieq 5407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑢( I ↾ 𝑋)𝑢 ↔ 𝑢 = 𝑢))
104	102, 103	mpbiri 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → 𝑢( I ↾ 𝑋)𝑢)
105	101, 101, 104	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑢 ∈ 𝐴) → 𝑢( I ↾ 𝑋)𝑢)
106		breq2 4657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑢 → (𝑢( I ↾ 𝑋)𝑣 ↔ 𝑢( I ↾ 𝑋)𝑢))
107	106	rspcev 3309	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢( I ↾ 𝑋)𝑢) → ∃𝑣 ∈ 𝐴 𝑢( I ↾ 𝑋)𝑣)
108	100, 105, 107	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑢 ∈ 𝐴) → ∃𝑣 ∈ 𝐴 𝑢( I ↾ 𝑋)𝑣)
109	108	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ 𝑋 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 𝑢( I ↾ 𝑋)𝑣)
110		dminxp 5574	. . . . . . . . . . . . . 14 ⊢ (dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 𝑢( I ↾ 𝑋)𝑣)
111	109, 110	sylibr 224	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = 𝐴)
112	111	reseq2d 5396	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 → ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴))) = ( I ↾ 𝐴))
113	99, 112	syl5req 2669	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑋 → ( I ↾ 𝐴) = (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
114	113	adantl 482	. . . . . . . . . 10 ⊢ ((( I ↾ 𝑋) ⊆ 𝑢 ∧ 𝐴 ⊆ 𝑋) → ( I ↾ 𝐴) = (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
115		ssrin 3838	. . . . . . . . . . 11 ⊢ (( I ↾ 𝑋) ⊆ 𝑢 → (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
116	115	adantr 481	. . . . . . . . . 10 ⊢ ((( I ↾ 𝑋) ⊆ 𝑢 ∧ 𝐴 ⊆ 𝑋) → (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
117	114, 116	eqsstrd 3639	. . . . . . . . 9 ⊢ ((( I ↾ 𝑋) ⊆ 𝑢 ∧ 𝐴 ⊆ 𝑋) → ( I ↾ 𝐴) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
118	93, 94, 117	syl2anc 693	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
119		simpr 477	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
120	118, 119	sseqtr4d 3642	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ 𝑣)
121	120, 57	r19.29a 3078	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ 𝑣)
122	15	ad3antrrr 766	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
123		ustinvel 22013	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ◡𝑢 ∈ 𝑈)
124	90, 91, 123	syl2anc 693	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ◡𝑢 ∈ 𝑈)
125	119	cnveqd 5298	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ◡𝑣 = ◡(𝑢 ∩ (𝐴 × 𝐴)))
126		cnvin 5540	. . . . . . . . . . 11 ⊢ ◡(𝑢 ∩ (𝐴 × 𝐴)) = (◡𝑢 ∩ ◡(𝐴 × 𝐴))
127		cnvxp 5551	. . . . . . . . . . . 12 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
128	127	ineq2i 3811	. . . . . . . . . . 11 ⊢ (◡𝑢 ∩ ◡(𝐴 × 𝐴)) = (◡𝑢 ∩ (𝐴 × 𝐴))
129	126, 128	eqtri 2644	. . . . . . . . . 10 ⊢ ◡(𝑢 ∩ (𝐴 × 𝐴)) = (◡𝑢 ∩ (𝐴 × 𝐴))
130	125, 129	syl6eq 2672	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ◡𝑣 = (◡𝑢 ∩ (𝐴 × 𝐴)))
131		ineq1 3807	. . . . . . . . . . 11 ⊢ (𝑥 = ◡𝑢 → (𝑥 ∩ (𝐴 × 𝐴)) = (◡𝑢 ∩ (𝐴 × 𝐴)))
132	131	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑥 = ◡𝑢 → (◡𝑣 = (𝑥 ∩ (𝐴 × 𝐴)) ↔ ◡𝑣 = (◡𝑢 ∩ (𝐴 × 𝐴))))
133	132	rspcev 3309	. . . . . . . . 9 ⊢ ((◡𝑢 ∈ 𝑈 ∧ ◡𝑣 = (◡𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 ◡𝑣 = (𝑥 ∩ (𝐴 × 𝐴)))
134	124, 130, 133	syl2anc 693	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥 ∈ 𝑈 ◡𝑣 = (𝑥 ∩ (𝐴 × 𝐴)))
135		elrest 16088	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝑈 ◡𝑣 = (𝑥 ∩ (𝐴 × 𝐴))))
136	135	biimpar 502	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑥 ∈ 𝑈 ◡𝑣 = (𝑥 ∩ (𝐴 × 𝐴))) → ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
137	90, 122, 134, 136	syl21anc 1325	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
138	137, 57	r19.29a 3078	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)))
139		simp-4l 806	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → 𝑈 ∈ (UnifOn‘𝑋))
140	15	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → (𝐴 × 𝐴) ∈ V)
141		simplr 792	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → 𝑥 ∈ 𝑈)
142		elrestr 16089	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑥 ∈ 𝑈) → (𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
143	139, 140, 141, 142	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → (𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
144		inss1 3833	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥
145		coss1 5277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ (𝑥 ∩ (𝐴 × 𝐴))))
146		coss2 5278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → (𝑥 ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ 𝑥))
147	145, 146	sstrd 3613	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ 𝑥))
148	144, 147	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ 𝑥)
149		sstr 3611	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ 𝑥) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
150	148, 149	mpan 706	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑢 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
151	150	adantl 482	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
152		inss2 3834	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
153		coss1 5277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝑥 ∩ (𝐴 × 𝐴))))
154		coss2 5278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)))
155	153, 154	sstrd 3613	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)))
156	152, 155	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))
157		xpidtr 5518	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
158	156, 157	sstri 3612	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
159	158	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))
160	151, 159	ssind 3837	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
161		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
162	161, 161	coeq12d 5286	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → (𝑤 ∘ 𝑤) = ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))))
163	162	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → ((𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)) ↔ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
164	163	rspcev 3309	. . . . . . . . . . 11 ⊢ (((𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
165	143, 160, 164	syl2anc 693	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) ∧ (𝑥 ∘ 𝑥) ⊆ 𝑢) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
166		ustexhalf 22014	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑢)
167	166	adantlr 751	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑥 ∘ 𝑥) ⊆ 𝑢)
168	165, 167	r19.29a 3078	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
169	168	ad4ant13 1292	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
170	119	sseq2d 3633	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
171	170	rexbidv 3052	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
172	169, 171	mpbird 247	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)
173	172, 57	r19.29a 3078	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)
174	121, 138, 173	3jca 1242	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣))
175	61, 89, 174	3jca 1242	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)))
176	175	ralrimiva 2966	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∀𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)))
177	2, 20, 176	3jca 1242	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝑈 ↾t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣))))
178		isust 22007	. . 3 ⊢ (𝐴 ∈ V → ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ↔ ((𝑈 ↾t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)))))
179	13, 178	syl 17	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ↔ ((𝑈 ↾t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣 ⊆ 𝑤 → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 ∩ 𝑤) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣 ∧ ◡𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑤 ∘ 𝑤) ⊆ 𝑣)))))
180	177, 179	mpbird 247	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158   class class class wbr 4653   I cid 5023   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  UnifOncust 22003

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-rep 4771  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-reu 2919  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-ust 22004

This theorem is referenced by:  restutop  22041  restutopopn  22042  ressust  22068  ressusp  22069  trcfilu  22098  cfiluweak  22099