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

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   o. 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