Theorem utopsnneiplem 22051

Description: The neighborhoods of a point P for the topology induced by an uniform space U. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypotheses

Ref	Expression
utoptop.1	|- J = (unifTop ` U )
utopsnneip.1	|- K = { a e. ~P X | A. p e. a a e. ( N ` p ) }
utopsnneip.2	|- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )

Assertion

Ref	Expression
utopsnneiplem	|- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` J ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )

Distinct variable groups:   p, a, K   N, a, p   v, p, P   v, a, U, p   X, a, p, v

Allowed substitution hints:   P( a)   J( v, p, a)   K( v)   N( v)

Proof of Theorem utopsnneiplem

Dummy variables b q u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		utoptop.1	. . . . . . . 8 |- J = (unifTop ` U )
2		utopval 22036	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = { a e. ~P X | A. p e. a E. w e. U ( w " { p } ) C_ a } )
3	1, 2	syl5eq 2668	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> J = { a e. ~P X | A. p e. a E. w e. U ( w " { p } ) C_ a } )
4		simpll 790	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) -> U e. (UnifOn ` X ) )
5		simpr 477	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) -> a e. ~P X )
6	5	elpwid 4170	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) -> a C_ X )
7	6	sselda 3603	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) -> p e. X )
8		simpr 477	. . . . . . . . . . . . . 14 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> p e. X )
9		mptexg 6484	. . . . . . . . . . . . . . . 16 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { p } ) ) e. _V )
10		rnexg 7098	. . . . . . . . . . . . . . . 16 |- ( ( v e. U |-> ( v " { p } ) ) e. _V -> ran ( v e. U |-> ( v " { p } ) ) e. _V )
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 |- ( U e. (UnifOn ` X ) -> ran ( v e. U |-> ( v " { p } ) ) e. _V )
12	11	adantr 481	. . . . . . . . . . . . . 14 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ran ( v e. U |-> ( v " { p } ) ) e. _V )
13		utopsnneip.2	. . . . . . . . . . . . . . 15 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
14	13	fvmpt2 6291	. . . . . . . . . . . . . 14 |- ( ( p e. X /\ ran ( v e. U |-> ( v " { p } ) ) e. _V ) -> ( N ` p ) = ran ( v e. U |-> ( v " { p } ) ) )
15	8, 12, 14	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( N ` p ) = ran ( v e. U |-> ( v " { p } ) ) )
16	15	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> a e. ran ( v e. U |-> ( v " { p } ) ) ) )
17		vex 3203	. . . . . . . . . . . . 13 |- a e. _V
18		eqid 2622	. . . . . . . . . . . . . 14 |- ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { p } ) )
19	18	elrnmpt 5372	. . . . . . . . . . . . 13 |- ( a e. _V -> ( a e. ran ( v e. U |-> ( v " { p } ) ) <-> E. v e. U a = ( v " { p } ) ) )
20	17, 19	ax-mp 5	. . . . . . . . . . . 12 |- ( a e. ran ( v e. U |-> ( v " { p } ) ) <-> E. v e. U a = ( v " { p } ) )
21	16, 20	syl6bb 276	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. v e. U a = ( v " { p } ) ) )
22	4, 7, 21	syl2anc 693	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) -> ( a e. ( N ` p ) <-> E. v e. U a = ( v " { p } ) ) )
23		nfv 1843	. . . . . . . . . . . . 13 |- F/ v ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a )
24		nfre1 3005	. . . . . . . . . . . . 13 |- F/ v E. v e. U a = ( v " { p } )
25	23, 24	nfan 1828	. . . . . . . . . . . 12 |- F/ v ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) )
26		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) ) /\ v e. U ) /\ a = ( v " { p } ) ) -> v e. U )
27		eqimss2 3658	. . . . . . . . . . . . . 14 |- ( a = ( v " { p } ) -> ( v " { p } ) C_ a )
28	27	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) ) /\ v e. U ) /\ a = ( v " { p } ) ) -> ( v " { p } ) C_ a )
29		imaeq1 5461	. . . . . . . . . . . . . . 15 |- ( w = v -> ( w " { p } ) = ( v " { p } ) )
30	29	sseq1d 3632	. . . . . . . . . . . . . 14 |- ( w = v -> ( ( w " { p } ) C_ a <-> ( v " { p } ) C_ a ) )
31	30	rspcev 3309	. . . . . . . . . . . . 13 |- ( ( v e. U /\ ( v " { p } ) C_ a ) -> E. w e. U ( w " { p } ) C_ a )
32	26, 28, 31	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) ) /\ v e. U ) /\ a = ( v " { p } ) ) -> E. w e. U ( w " { p } ) C_ a )
33		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) ) -> E. v e. U a = ( v " { p } ) )
34	25, 32, 33	r19.29af 3076	. . . . . . . . . . 11 |- ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. v e. U a = ( v " { p } ) ) -> E. w e. U ( w " { p } ) C_ a )
35	4	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> U e. (UnifOn ` X ) )
36	7	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> p e. X )
37	35, 36	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> ( U e. (UnifOn ` X ) /\ p e. X ) )
38		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> ( w " { p } ) C_ a )
39	6	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> a C_ X )
40		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> w e. U )
41		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( w " { p } ) = ( w " { p } )
42		imaeq1 5461	. . . . . . . . . . . . . . . . . . . 20 |- ( u = w -> ( u " { p } ) = ( w " { p } ) )
43	42	eqeq2d 2632	. . . . . . . . . . . . . . . . . . 19 |- ( u = w -> ( ( w " { p } ) = ( u " { p } ) <-> ( w " { p } ) = ( w " { p } ) ) )
44	43	rspcev 3309	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. U /\ ( w " { p } ) = ( w " { p } ) ) -> E. u e. U ( w " { p } ) = ( u " { p } ) )
45	41, 44	mpan2 707	. . . . . . . . . . . . . . . . 17 |- ( w e. U -> E. u e. U ( w " { p } ) = ( u " { p } ) )
46	45	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ w e. U ) -> E. u e. U ( w " { p } ) = ( u " { p } ) )
47		vex 3203	. . . . . . . . . . . . . . . . . . 19 |- w e. _V
48	47	imaex 7104	. . . . . . . . . . . . . . . . . 18 |- ( w " { p } ) e. _V
49	13	ustuqtoplem 22043	. . . . . . . . . . . . . . . . . 18 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( w " { p } ) e. _V ) -> ( ( w " { p } ) e. ( N ` p ) <-> E. u e. U ( w " { p } ) = ( u " { p } ) ) )
50	48, 49	mpan2 707	. . . . . . . . . . . . . . . . 17 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( ( w " { p } ) e. ( N ` p ) <-> E. u e. U ( w " { p } ) = ( u " { p } ) ) )
51	50	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ w e. U ) -> ( ( w " { p } ) e. ( N ` p ) <-> E. u e. U ( w " { p } ) = ( u " { p } ) ) )
52	46, 51	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ w e. U ) -> ( w " { p } ) e. ( N ` p ) )
53	35, 36, 40, 52	syl21anc 1325	. . . . . . . . . . . . . 14 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> ( w " { p } ) e. ( N ` p ) )
54		sseq1 3626	. . . . . . . . . . . . . . . . . 18 |- ( b = ( w " { p } ) -> ( b C_ a <-> ( w " { p } ) C_ a ) )
55	54	3anbi2d 1404	. . . . . . . . . . . . . . . . 17 |- ( b = ( w " { p } ) -> ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b C_ a /\ a C_ X ) <-> ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( w " { p } ) C_ a /\ a C_ X ) ) )
56		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( b = ( w " { p } ) -> ( b e. ( N ` p ) <-> ( w " { p } ) e. ( N ` p ) ) )
57	55, 56	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( b = ( w " { p } ) -> ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b C_ a /\ a C_ X ) /\ b e. ( N ` p ) ) <-> ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( w " { p } ) C_ a /\ a C_ X ) /\ ( w " { p } ) e. ( N ` p ) ) ) )
58	57	imbi1d 331	. . . . . . . . . . . . . . 15 |- ( b = ( w " { p } ) -> ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b C_ a /\ a C_ X ) /\ b e. ( N ` p ) ) -> a e. ( N ` p ) ) <-> ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( w " { p } ) C_ a /\ a C_ X ) /\ ( w " { p } ) e. ( N ` p ) ) -> a e. ( N ` p ) ) ) )
59	13	ustuqtop1 22045	. . . . . . . . . . . . . . 15 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b C_ a /\ a C_ X ) /\ b e. ( N ` p ) ) -> a e. ( N ` p ) )
60	48, 58, 59	vtocl 3259	. . . . . . . . . . . . . 14 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( w " { p } ) C_ a /\ a C_ X ) /\ ( w " { p } ) e. ( N ` p ) ) -> a e. ( N ` p ) )
61	37, 38, 39, 53, 60	syl31anc 1329	. . . . . . . . . . . . 13 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> a e. ( N ` p ) )
62	37, 21	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> ( a e. ( N ` p ) <-> E. v e. U a = ( v " { p } ) ) )
63	61, 62	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ w e. U ) /\ ( w " { p } ) C_ a ) -> E. v e. U a = ( v " { p } ) )
64	63	r19.29an 3077	. . . . . . . . . . 11 |- ( ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) /\ E. w e. U ( w " { p } ) C_ a ) -> E. v e. U a = ( v " { p } ) )
65	34, 64	impbida 877	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) -> ( E. v e. U a = ( v " { p } ) <-> E. w e. U ( w " { p } ) C_ a ) )
66	22, 65	bitrd 268	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) /\ p e. a ) -> ( a e. ( N ` p ) <-> E. w e. U ( w " { p } ) C_ a ) )
67	66	ralbidva 2985	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ a e. ~P X ) -> ( A. p e. a a e. ( N ` p ) <-> A. p e. a E. w e. U ( w " { p } ) C_ a ) )
68	67	rabbidva 3188	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> { a e. ~P X | A. p e. a a e. ( N ` p ) } = { a e. ~P X | A. p e. a E. w e. U ( w " { p } ) C_ a } )
69	3, 68	eqtr4d 2659	. . . . . 6 |- ( U e. (UnifOn ` X ) -> J = { a e. ~P X | A. p e. a a e. ( N ` p ) } )
70		utopsnneip.1	. . . . . 6 |- K = { a e. ~P X | A. p e. a a e. ( N ` p ) }
71	69, 70	syl6eqr 2674	. . . . 5 |- ( U e. (UnifOn ` X ) -> J = K )
72	71	fveq2d 6195	. . . 4 |- ( U e. (UnifOn ` X ) -> ( nei ` J ) = ( nei ` K ) )
73	72	fveq1d 6193	. . 3 |- ( U e. (UnifOn ` X ) -> ( ( nei ` J ) ` { P } ) = ( ( nei ` K ) ` { P } ) )
74	73	adantr 481	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` J ) ` { P } ) = ( ( nei ` K ) ` { P } ) )
75	13	ustuqtop0 22044	. . . . 5 |- ( U e. (UnifOn ` X ) -> N : X --> ~P ~P X )
76	13	ustuqtop1 22045	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )
77	13	ustuqtop2 22046	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
78	13	ustuqtop3 22047	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )
79	13	ustuqtop4 22048	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. b e. ( N ` p ) A. q e. b a e. ( N ` q ) )
80	13	ustuqtop5 22049	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> X e. ( N ` p ) )
81	70, 75, 76, 77, 78, 79, 80	neiptopnei 20936	. . . 4 |- ( U e. (UnifOn ` X ) -> N = ( p e. X |-> ( ( nei ` K ) ` { p } ) ) )
82	81	adantr 481	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> N = ( p e. X |-> ( ( nei ` K ) ` { p } ) ) )
83		simpr 477	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ p = P ) -> p = P )
84	83	sneqd 4189	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ p = P ) -> { p } = { P } )
85	84	fveq2d 6195	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ p = P ) -> ( ( nei ` K ) ` { p } ) = ( ( nei ` K ) ` { P } ) )
86		simpr 477	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> P e. X )
87		fvexd 6203	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` K ) ` { P } ) e. _V )
88	82, 85, 86, 87	fvmptd 6288	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ( ( nei ` K ) ` { P } ) )
89		mptexg 6484	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { P } ) ) e. _V )
90		rnexg 7098	. . . . 5 |- ( ( v e. U |-> ( v " { P } ) ) e. _V -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
91	89, 90	syl 17	. . . 4 |- ( U e. (UnifOn ` X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
92	91	adantr 481	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
93	13	a1i 11	. . . 4 |- ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) -> N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) )
94		nfv 1843	. . . . . . . 8 |- F/ v P e. X
95		nfmpt1 4747	. . . . . . . . . 10 |- F/_ v ( v e. U |-> ( v " { P } ) )
96	95	nfrn 5368	. . . . . . . . 9 |- F/_ v ran ( v e. U |-> ( v " { P } ) )
97	96	nfel1 2779	. . . . . . . 8 |- F/ v ran ( v e. U |-> ( v " { P } ) ) e. _V
98	94, 97	nfan 1828	. . . . . . 7 |- F/ v ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V )
99		nfv 1843	. . . . . . 7 |- F/ v p = P
100	98, 99	nfan 1828	. . . . . 6 |- F/ v ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) /\ p = P )
101		simpr2 1068	. . . . . . . . 9 |- ( ( P e. X /\ ( ran ( v e. U |-> ( v " { P } ) ) e. _V /\ p = P /\ v e. U ) ) -> p = P )
102	101	sneqd 4189	. . . . . . . 8 |- ( ( P e. X /\ ( ran ( v e. U |-> ( v " { P } ) ) e. _V /\ p = P /\ v e. U ) ) -> { p } = { P } )
103	102	imaeq2d 5466	. . . . . . 7 |- ( ( P e. X /\ ( ran ( v e. U |-> ( v " { P } ) ) e. _V /\ p = P /\ v e. U ) ) -> ( v " { p } ) = ( v " { P } ) )
104	103	3anassrs 1290	. . . . . 6 |- ( ( ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) /\ p = P ) /\ v e. U ) -> ( v " { p } ) = ( v " { P } ) )
105	100, 104	mpteq2da 4743	. . . . 5 |- ( ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) /\ p = P ) -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { P } ) ) )
106	105	rneqd 5353	. . . 4 |- ( ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) /\ p = P ) -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { P } ) ) )
107		simpl 473	. . . 4 |- ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) -> P e. X )
108		simpr 477	. . . 4 |- ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
109	93, 106, 107, 108	fvmptd 6288	. . 3 |- ( ( P e. X /\ ran ( v e. U |-> ( v " { P } ) ) e. _V ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
110	86, 92, 109	syl2anc 693	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
111	74, 88, 110	3eqtr2d 2662	1 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` J ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ran crn 5115   "cima 5117   ` cfv 5888   neicnei 20901  UnifOncust 22003  unifTopcutop 22034

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-top 20699  df-nei 20902  df-ust 22004  df-utop 22035

This theorem is referenced by:  utopsnneip  22052