Theorem metucn 20163

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 20117. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypotheses

Ref	Expression
metucn.u	|- U = (metUnif ` C )
metucn.v	|- V = (metUnif ` D )
metucn.x	|- ( ph -> X =/= (/) )
metucn.y	|- ( ph -> Y =/= (/) )
metucn.c	|- ( ph -> C e. (PsMet ` X ) )
metucn.d	|- ( ph -> D e. (PsMet ` Y ) )

Assertion

Ref	Expression
metucn	|- ( ph -> ( F e. ( U Cnu V ) <-> ( F : X --> Y /\ A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) ) )

Distinct variable groups:   c, d, x, y, C   D, c, d, x, y   F, c, d, x, y   x, U, y   x, V   X, c, d, x, y   Y, c, d, x, y   ph, c, d, x, y

Allowed substitution hints:   U( c, d)   V( y, c, d)

Proof of Theorem metucn

Dummy variables a e u v b f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metucn.u	. . . . . 6 |- U = (metUnif ` C )
2		metucn.c	. . . . . . 7 |- ( ph -> C e. (PsMet ` X ) )
3		metuval 20124	. . . . . . 7 |- ( C e. (PsMet ` X ) -> (metUnif ` C ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) )
4	2, 3	syl 16	. . . . . 6 |- ( ph -> (metUnif ` C ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) )
5	1, 4	syl5eq 2486	. . . . 5 |- ( ph -> U = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) )
6		metucn.v	. . . . . 6 |- V = (metUnif ` D )
7		metucn.d	. . . . . . 7 |- ( ph -> D e. (PsMet ` Y ) )
8		metuval 20124	. . . . . . 7 |- ( D e. (PsMet ` Y ) -> (metUnif ` D ) = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
9	7, 8	syl 16	. . . . . 6 |- ( ph -> (metUnif ` D ) = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
10	6, 9	syl5eq 2486	. . . . 5 |- ( ph -> V = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
11	5, 10	oveq12d 6108	. . . 4 |- ( ph -> ( U Cnu V ) = ( ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) Cnu ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) ) )
12	11	eleq2d 2509	. . 3 |- ( ph -> ( F e. ( U Cnu V ) <-> F e. ( ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) Cnu ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) ) ) )
13		eqid 2442	. . . 4 |- ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) )
14		eqid 2442	. . . 4 |- ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) )
15		metucn.x	. . . . 5 |- ( ph -> X =/= (/) )
16		oveq2 6098	. . . . . . . . 9 |- ( a = c -> ( 0 [,) a ) = ( 0 [,) c ) )
17	16	imaeq2d 5168	. . . . . . . 8 |- ( a = c -> ( `' C " ( 0 [,) a ) ) = ( `' C " ( 0 [,) c ) ) )
18	17	cbvmptv 4382	. . . . . . 7 |- ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ( c e. RR+ |-> ( `' C " ( 0 [,) c ) ) )
19	18	rneqi 5065	. . . . . 6 |- ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ran ( c e. RR+ |-> ( `' C " ( 0 [,) c ) ) )
20	19	metust 20142	. . . . 5 |- ( ( X =/= (/) /\ C e. (PsMet ` X ) ) -> ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) e. (UnifOn ` X ) )
21	15, 2, 20	syl2anc 661	. . . 4 |- ( ph -> ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) e. (UnifOn ` X ) )
22		metucn.y	. . . . 5 |- ( ph -> Y =/= (/) )
23		oveq2 6098	. . . . . . . . 9 |- ( b = d -> ( 0 [,) b ) = ( 0 [,) d ) )
24	23	imaeq2d 5168	. . . . . . . 8 |- ( b = d -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) d ) ) )
25	24	cbvmptv 4382	. . . . . . 7 |- ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
26	25	rneqi 5065	. . . . . 6 |- ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
27	26	metust 20142	. . . . 5 |- ( ( Y =/= (/) /\ D e. (PsMet ` Y ) ) -> ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) e. (UnifOn ` Y ) )
28	22, 7, 27	syl2anc 661	. . . 4 |- ( ph -> ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) e. (UnifOn ` Y ) )
29		oveq2 6098	. . . . . . . . 9 |- ( a = e -> ( 0 [,) a ) = ( 0 [,) e ) )
30	29	imaeq2d 5168	. . . . . . . 8 |- ( a = e -> ( `' C " ( 0 [,) a ) ) = ( `' C " ( 0 [,) e ) ) )
31	30	cbvmptv 4382	. . . . . . 7 |- ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ( e e. RR+ |-> ( `' C " ( 0 [,) e ) ) )
32	31	rneqi 5065	. . . . . 6 |- ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ran ( e e. RR+ |-> ( `' C " ( 0 [,) e ) ) )
33	32	metustfbas 20140	. . . . 5 |- ( ( X =/= (/) /\ C e. (PsMet ` X ) ) -> ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) e. ( fBas ` ( X X. X ) ) )
34	15, 2, 33	syl2anc 661	. . . 4 |- ( ph -> ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) e. ( fBas ` ( X X. X ) ) )
35		oveq2 6098	. . . . . . . . 9 |- ( b = f -> ( 0 [,) b ) = ( 0 [,) f ) )
36	35	imaeq2d 5168	. . . . . . . 8 |- ( b = f -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) f ) ) )
37	36	cbvmptv 4382	. . . . . . 7 |- ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ( f e. RR+ |-> ( `' D " ( 0 [,) f ) ) )
38	37	rneqi 5065	. . . . . 6 |- ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ran ( f e. RR+ |-> ( `' D " ( 0 [,) f ) ) )
39	38	metustfbas 20140	. . . . 5 |- ( ( Y =/= (/) /\ D e. (PsMet ` Y ) ) -> ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) e. ( fBas ` ( Y X. Y ) ) )
40	22, 7, 39	syl2anc 661	. . . 4 |- ( ph -> ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) e. ( fBas ` ( Y X. Y ) ) )
41	13, 14, 21, 28, 34, 40	isucn2 19853	. . 3 |- ( ph -> ( F e. ( ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) Cnu ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) ) <-> ( F : X --> Y /\ A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) ) ) )
42	12, 41	bitrd 253	. 2 |- ( ph -> ( F e. ( U Cnu V ) <-> ( F : X --> Y /\ A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) ) ) )
43		eqid 2442	. . . . . . . . . 10 |- ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) )
44		oveq2 6098	. . . . . . . . . . . . 13 |- ( f = d -> ( 0 [,) f ) = ( 0 [,) d ) )
45	44	imaeq2d 5168	. . . . . . . . . . . 12 |- ( f = d -> ( `' D " ( 0 [,) f ) ) = ( `' D " ( 0 [,) d ) ) )
46	45	eqeq2d 2453	. . . . . . . . . . 11 |- ( f = d -> ( ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) <-> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) ) ) )
47	46	rspcev 3072	. . . . . . . . . 10 |- ( ( d e. RR+ /\ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) ) ) -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
48	43, 47	mpan2 671	. . . . . . . . 9 |- ( d e. RR+ -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
49	48	adantl 466	. . . . . . . 8 |- ( ( ph /\ d e. RR+ ) -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
50	38	metustel 20126	. . . . . . . . . 10 |- ( D e. (PsMet ` Y ) -> ( ( `' D " ( 0 [,) d ) ) e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) ) )
51	7, 50	syl 16	. . . . . . . . 9 |- ( ph -> ( ( `' D " ( 0 [,) d ) ) e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) ) )
52	51	adantr 465	. . . . . . . 8 |- ( ( ph /\ d e. RR+ ) -> ( ( `' D " ( 0 [,) d ) ) e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) ) )
53	49, 52	mpbird 232	. . . . . . 7 |- ( ( ph /\ d e. RR+ ) -> ( `' D " ( 0 [,) d ) ) e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) )
54	26	metustel 20126	. . . . . . . . 9 |- ( D e. (PsMet ` Y ) -> ( v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. d e. RR+ v = ( `' D " ( 0 [,) d ) ) ) )
55	7, 54	syl 16	. . . . . . . 8 |- ( ph -> ( v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. d e. RR+ v = ( `' D " ( 0 [,) d ) ) ) )
56	55	biimpa 484	. . . . . . 7 |- ( ( ph /\ v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) -> E. d e. RR+ v = ( `' D " ( 0 [,) d ) ) )
57		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> v = ( `' D " ( 0 [,) d ) ) )
58	57	breqd 4302	. . . . . . . . . 10 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( ( F ` x ) v ( F ` y ) <-> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) )
59	58	imbi2d 316	. . . . . . . . 9 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( ( x u y -> ( F ` x ) v ( F ` y ) ) <-> ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
60	59	ralbidv 2734	. . . . . . . 8 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
61	60	rexralbidv 2758	. . . . . . 7 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
62	53, 56, 61	ralxfrd 4505	. . . . . 6 |- ( ph -> ( A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> A. d e. RR+ E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
63		eqid 2442	. . . . . . . . . . 11 |- ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) )
64		oveq2 6098	. . . . . . . . . . . . . 14 |- ( e = c -> ( 0 [,) e ) = ( 0 [,) c ) )
65	64	imaeq2d 5168	. . . . . . . . . . . . 13 |- ( e = c -> ( `' C " ( 0 [,) e ) ) = ( `' C " ( 0 [,) c ) ) )
66	65	eqeq2d 2453	. . . . . . . . . . . 12 |- ( e = c -> ( ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) <-> ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) ) ) )
67	66	rspcev 3072	. . . . . . . . . . 11 |- ( ( c e. RR+ /\ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) ) ) -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
68	63, 67	mpan2 671	. . . . . . . . . 10 |- ( c e. RR+ -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
69	68	adantl 466	. . . . . . . . 9 |- ( ( ph /\ c e. RR+ ) -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
70	32	metustel 20126	. . . . . . . . . . 11 |- ( C e. (PsMet ` X ) -> ( ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) ) )
71	2, 70	syl 16	. . . . . . . . . 10 |- ( ph -> ( ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) ) )
72	71	adantr 465	. . . . . . . . 9 |- ( ( ph /\ c e. RR+ ) -> ( ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) ) )
73	69, 72	mpbird 232	. . . . . . . 8 |- ( ( ph /\ c e. RR+ ) -> ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) )
74	19	metustel 20126	. . . . . . . . . 10 |- ( C e. (PsMet ` X ) -> ( u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. c e. RR+ u = ( `' C " ( 0 [,) c ) ) ) )
75	2, 74	syl 16	. . . . . . . . 9 |- ( ph -> ( u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. c e. RR+ u = ( `' C " ( 0 [,) c ) ) ) )
76	75	biimpa 484	. . . . . . . 8 |- ( ( ph /\ u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) -> E. c e. RR+ u = ( `' C " ( 0 [,) c ) ) )
77		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> u = ( `' C " ( 0 [,) c ) ) )
78	77	breqd 4302	. . . . . . . . . 10 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> ( x u y <-> x ( `' C " ( 0 [,) c ) ) y ) )
79	78	imbi1d 317	. . . . . . . . 9 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> ( ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
80	79	2ralbidv 2756	. . . . . . . 8 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> ( A. x e. X A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
81	73, 76, 80	rexxfrd 4506	. . . . . . 7 |- ( ph -> ( E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
82	81	ralbidv 2734	. . . . . 6 |- ( ph -> ( A. d e. RR+ E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
83	62, 82	bitrd 253	. . . . 5 |- ( ph -> ( A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
84	83	adantr 465	. . . 4 |- ( ( ph /\ F : X --> Y ) -> ( A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
85	2	ad4antr 731	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> C e. (PsMet ` X ) )
86		simplr 754	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> c e. RR+ )
87		simprr 756	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
88		simprl 755	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
89		elbl4 20150	. . . . . . . . . 10 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR+ ) /\ ( y e. X /\ x e. X ) ) -> ( x e. ( y ( ball ` C ) c ) <-> x ( `' C " ( 0 [,) c ) ) y ) )
90		rpxr 10997	. . . . . . . . . . 11 |- ( c e. RR+ -> c e. RR* )
91		elbl3ps 19965	. . . . . . . . . . 11 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR* ) /\ ( y e. X /\ x e. X ) ) -> ( x e. ( y ( ball ` C ) c ) <-> ( x C y ) < c ) )
92	90, 91	sylanl2 651	. . . . . . . . . 10 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR+ ) /\ ( y e. X /\ x e. X ) ) -> ( x e. ( y ( ball ` C ) c ) <-> ( x C y ) < c ) )
93	89, 92	bitr3d 255	. . . . . . . . 9 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR+ ) /\ ( y e. X /\ x e. X ) ) -> ( x ( `' C " ( 0 [,) c ) ) y <-> ( x C y ) < c ) )
94	85, 86, 87, 88, 93	syl22anc 1219	. . . . . . . 8 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( x ( `' C " ( 0 [,) c ) ) y <-> ( x C y ) < c ) )
95	7	ad4antr 731	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> D e. (PsMet ` Y ) )
96		simpllr 758	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> d e. RR+ )
97		simp-4r 766	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> F : X --> Y )
98	97, 87	ffvelrnd 5843	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( F ` y ) e. Y )
99	97, 88	ffvelrnd 5843	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( F ` x ) e. Y )
100		elbl4 20150	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` Y ) /\ d e. RR+ ) /\ ( ( F ` y ) e. Y /\ ( F ` x ) e. Y ) ) -> ( ( F ` x ) e. ( ( F ` y ) ( ball ` D ) d ) <-> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) )
101		rpxr 10997	. . . . . . . . . . 11 |- ( d e. RR+ -> d e. RR* )
102		elbl3ps 19965	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` Y ) /\ d e. RR* ) /\ ( ( F ` y ) e. Y /\ ( F ` x ) e. Y ) ) -> ( ( F ` x ) e. ( ( F ` y ) ( ball ` D ) d ) <-> ( ( F ` x ) D ( F ` y ) ) < d ) )
103	101, 102	sylanl2 651	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` Y ) /\ d e. RR+ ) /\ ( ( F ` y ) e. Y /\ ( F ` x ) e. Y ) ) -> ( ( F ` x ) e. ( ( F ` y ) ( ball ` D ) d ) <-> ( ( F ` x ) D ( F ` y ) ) < d ) )
104	100, 103	bitr3d 255	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` Y ) /\ d e. RR+ ) /\ ( ( F ` y ) e. Y /\ ( F ` x ) e. Y ) ) -> ( ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) <-> ( ( F ` x ) D ( F ` y ) ) < d ) )
105	95, 96, 98, 99, 104	syl22anc 1219	. . . . . . . 8 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) <-> ( ( F ` x ) D ( F ` y ) ) < d ) )
106	94, 105	imbi12d 320	. . . . . . 7 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) )
107	106	2ralbidva 2754	. . . . . 6 |- ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) -> ( A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) )
108	107	rexbidva 2731	. . . . 5 |- ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) -> ( E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) )
109	108	ralbidva 2730	. . . 4 |- ( ( ph /\ F : X --> Y ) -> ( A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( x ( `' C " ( 0 [,) c ) ) y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) <-> A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) )
110	84, 109	bitrd 253	. . 3 |- ( ( ph /\ F : X --> Y ) -> ( A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) <-> A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) )
111	110	pm5.32da 641	. 2 |- ( ph -> ( ( F : X --> Y /\ A. v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) E. u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) A. x e. X A. y e. X ( x u y -> ( F ` x ) v ( F ` y ) ) ) <-> ( F : X --> Y /\ A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) ) )
112	42, 111	bitrd 253	1 |- ( ph -> ( F e. ( U Cnu V ) <-> ( F : X --> Y /\ A. d e. RR+ E. c e. RR+ A. x e. X A. y e. X ( ( x C y ) < c -> ( ( F ` x ) D ( F ` y ) ) < d ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2605   A.wral 2714   E.wrex 2715   (/)c0 3636   class class class wbr 4291   e. cmpt 4349   X. cxp 4837   `'ccnv 4838   ran crn 4840   "cima 4842   -->wf 5413   ` cfv 5417  (class class class)co 6090   0cc0 9281   RR*cxr 9416   < clt 9417   RR+crp 10990   [,)cico 11301  PsMetcpsmet 17799   ballcbl 17802   fBascfbas 17803   filGencfg 17804  metUnifcmetu 17807  UnifOncust 19773   Cn_ucucn 19849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-2 10379  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ico 11305  df-psmet 17808  df-bl 17811  df-fbas 17813  df-fg 17814  df-metu 17816  df-fil 19418  df-ust 19774  df-ucn 19850

This theorem is referenced by:  qqhucn  26420  heicant  28424