Theorem metucn 21517

Description: Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 21489. (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 21495	. . . . . . 7 |- ( C e. (PsMet ` X ) -> (metUnif ` C ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) )
4	2, 3	syl 17	. . . . . 6 |- ( ph -> (metUnif ` C ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) )
5	1, 4	syl5eq 2482	. . . . 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 21495	. . . . . . 7 |- ( D e. (PsMet ` Y ) -> (metUnif ` D ) = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
9	7, 8	syl 17	. . . . . 6 |- ( ph -> (metUnif ` D ) = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
10	6, 9	syl5eq 2482	. . . . 5 |- ( ph -> V = ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) )
11	5, 10	oveq12d 6323	. . . 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 2499	. . 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 2429	. . . 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 2429	. . . 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 6313	. . . . . . . . 9 |- ( a = c -> ( 0 [,) a ) = ( 0 [,) c ) )
17	16	imaeq2d 5188	. . . . . . . 8 |- ( a = c -> ( `' C " ( 0 [,) a ) ) = ( `' C " ( 0 [,) c ) ) )
18	17	cbvmptv 4518	. . . . . . 7 |- ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ( c e. RR+ |-> ( `' C " ( 0 [,) c ) ) )
19	18	rneqi 5081	. . . . . 6 |- ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ran ( c e. RR+ |-> ( `' C " ( 0 [,) c ) ) )
20	19	metust 21504	. . . . 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 665	. . . 4 |- ( ph -> ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) ) e. (UnifOn ` X ) )
22		metucn.y	. . . . 5 |- ( ph -> Y =/= (/) )
23		oveq2 6313	. . . . . . . . 9 |- ( b = d -> ( 0 [,) b ) = ( 0 [,) d ) )
24	23	imaeq2d 5188	. . . . . . . 8 |- ( b = d -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) d ) ) )
25	24	cbvmptv 4518	. . . . . . 7 |- ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
26	25	rneqi 5081	. . . . . 6 |- ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
27	26	metust 21504	. . . . 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 665	. . . 4 |- ( ph -> ( ( Y X. Y ) filGen ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) ) e. (UnifOn ` Y ) )
29		oveq2 6313	. . . . . . . . 9 |- ( a = e -> ( 0 [,) a ) = ( 0 [,) e ) )
30	29	imaeq2d 5188	. . . . . . . 8 |- ( a = e -> ( `' C " ( 0 [,) a ) ) = ( `' C " ( 0 [,) e ) ) )
31	30	cbvmptv 4518	. . . . . . 7 |- ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ( e e. RR+ |-> ( `' C " ( 0 [,) e ) ) )
32	31	rneqi 5081	. . . . . 6 |- ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) = ran ( e e. RR+ |-> ( `' C " ( 0 [,) e ) ) )
33	32	metustfbas 21503	. . . . 5 |- ( ( X =/= (/) /\ C e. (PsMet ` X ) ) -> ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) e. ( fBas ` ( X X. X ) ) )
34	15, 2, 33	syl2anc 665	. . . 4 |- ( ph -> ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) e. ( fBas ` ( X X. X ) ) )
35		oveq2 6313	. . . . . . . . 9 |- ( b = f -> ( 0 [,) b ) = ( 0 [,) f ) )
36	35	imaeq2d 5188	. . . . . . . 8 |- ( b = f -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) f ) ) )
37	36	cbvmptv 4518	. . . . . . 7 |- ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ( f e. RR+ |-> ( `' D " ( 0 [,) f ) ) )
38	37	rneqi 5081	. . . . . 6 |- ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) = ran ( f e. RR+ |-> ( `' D " ( 0 [,) f ) ) )
39	38	metustfbas 21503	. . . . 5 |- ( ( Y =/= (/) /\ D e. (PsMet ` Y ) ) -> ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) e. ( fBas ` ( Y X. Y ) ) )
40	22, 7, 39	syl2anc 665	. . . 4 |- ( ph -> ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) e. ( fBas ` ( Y X. Y ) ) )
41	13, 14, 21, 28, 34, 40	isucn2 21225	. . 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 256	. 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 2429	. . . . . . . . . 10 |- ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) )
44		oveq2 6313	. . . . . . . . . . . . 13 |- ( f = d -> ( 0 [,) f ) = ( 0 [,) d ) )
45	44	imaeq2d 5188	. . . . . . . . . . . 12 |- ( f = d -> ( `' D " ( 0 [,) f ) ) = ( `' D " ( 0 [,) d ) ) )
46	45	eqeq2d 2443	. . . . . . . . . . 11 |- ( f = d -> ( ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) <-> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) ) ) )
47	46	rspcev 3188	. . . . . . . . . 10 |- ( ( d e. RR+ /\ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) d ) ) ) -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
48	43, 47	mpan2 675	. . . . . . . . 9 |- ( d e. RR+ -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
49	48	adantl 467	. . . . . . . 8 |- ( ( ph /\ d e. RR+ ) -> E. f e. RR+ ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) f ) ) )
50	38	metustel 21496	. . . . . . . . . 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 17	. . . . . . . . 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 466	. . . . . . . 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 235	. . . . . . 7 |- ( ( ph /\ d e. RR+ ) -> ( `' D " ( 0 [,) d ) ) e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) )
54	26	metustel 21496	. . . . . . . 8 |- ( 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 17	. . . . . . 7 |- ( ph -> ( v e. ran ( b e. RR+ |-> ( `' D " ( 0 [,) b ) ) ) <-> E. d e. RR+ v = ( `' D " ( 0 [,) d ) ) ) )
56		simpr 462	. . . . . . . . . . 11 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> v = ( `' D " ( 0 [,) d ) ) )
57	56	breqd 4437	. . . . . . . . . 10 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( ( F ` x ) v ( F ` y ) <-> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) )
58	57	imbi2d 317	. . . . . . . . 9 |- ( ( ph /\ v = ( `' D " ( 0 [,) d ) ) ) -> ( ( x u y -> ( F ` x ) v ( F ` y ) ) <-> ( x u y -> ( F ` x ) ( `' D " ( 0 [,) d ) ) ( F ` y ) ) ) )
59	58	ralbidv 2871	. . . . . . . 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 ) ) ) )
60	59	rexralbidv 2954	. . . . . . 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 ) ) ) )
61	53, 55, 60	ralxfr2d 4638	. . . . . 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 ) ) ) )
62		eqid 2429	. . . . . . . . . . 11 |- ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) )
63		oveq2 6313	. . . . . . . . . . . . . 14 |- ( e = c -> ( 0 [,) e ) = ( 0 [,) c ) )
64	63	imaeq2d 5188	. . . . . . . . . . . . 13 |- ( e = c -> ( `' C " ( 0 [,) e ) ) = ( `' C " ( 0 [,) c ) ) )
65	64	eqeq2d 2443	. . . . . . . . . . . 12 |- ( e = c -> ( ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) <-> ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) ) ) )
66	65	rspcev 3188	. . . . . . . . . . 11 |- ( ( c e. RR+ /\ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) c ) ) ) -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
67	62, 66	mpan2 675	. . . . . . . . . 10 |- ( c e. RR+ -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
68	67	adantl 467	. . . . . . . . 9 |- ( ( ph /\ c e. RR+ ) -> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) )
69	32	metustel 21496	. . . . . . . . . . 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 ) ) ) )
70	2, 69	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. e e. RR+ ( `' C " ( 0 [,) c ) ) = ( `' C " ( 0 [,) e ) ) ) )
71	70	adantr 466	. . . . . . . . 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 ) ) ) )
72	68, 71	mpbird 235	. . . . . . . 8 |- ( ( ph /\ c e. RR+ ) -> ( `' C " ( 0 [,) c ) ) e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) )
73	19	metustel 21496	. . . . . . . . 9 |- ( C e. (PsMet ` X ) -> ( u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. c e. RR+ u = ( `' C " ( 0 [,) c ) ) ) )
74	2, 73	syl 17	. . . . . . . 8 |- ( ph -> ( u e. ran ( a e. RR+ |-> ( `' C " ( 0 [,) a ) ) ) <-> E. c e. RR+ u = ( `' C " ( 0 [,) c ) ) ) )
75		simpr 462	. . . . . . . . . . 11 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> u = ( `' C " ( 0 [,) c ) ) )
76	75	breqd 4437	. . . . . . . . . 10 |- ( ( ph /\ u = ( `' C " ( 0 [,) c ) ) ) -> ( x u y <-> x ( `' C " ( 0 [,) c ) ) y ) )
77	76	imbi1d 318	. . . . . . . . 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 ) ) ) )
78	77	2ralbidv 2876	. . . . . . . 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 ) ) ) )
79	72, 74, 78	rexxfr2d 4639	. . . . . . 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 ) ) ) )
80	79	ralbidv 2871	. . . . . 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 ) ) ) )
81	61, 80	bitrd 256	. . . . 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 ) ) ) )
82	81	adantr 466	. . . 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 ) ) ) )
83	2	ad4antr 736	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> C e. (PsMet ` X ) )
84		simplr 760	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> c e. RR+ )
85		simprr 764	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
86		simprl 762	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
87		elbl4 21509	. . . . . . . . . 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 ) )
88		rpxr 11309	. . . . . . . . . . 11 |- ( c e. RR+ -> c e. RR* )
89		elbl3ps 21337	. . . . . . . . . . 11 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR* ) /\ ( y e. X /\ x e. X ) ) -> ( x e. ( y ( ball ` C ) c ) <-> ( x C y ) < c ) )
90	88, 89	sylanl2 655	. . . . . . . . . 10 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR+ ) /\ ( y e. X /\ x e. X ) ) -> ( x e. ( y ( ball ` C ) c ) <-> ( x C y ) < c ) )
91	87, 90	bitr3d 258	. . . . . . . . 9 |- ( ( ( C e. (PsMet ` X ) /\ c e. RR+ ) /\ ( y e. X /\ x e. X ) ) -> ( x ( `' C " ( 0 [,) c ) ) y <-> ( x C y ) < c ) )
92	83, 84, 85, 86, 91	syl22anc 1265	. . . . . . . 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 ) )
93	7	ad4antr 736	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> D e. (PsMet ` Y ) )
94		simpllr 767	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> d e. RR+ )
95		simp-4r 775	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> F : X --> Y )
96	95, 85	ffvelrnd 6038	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( F ` y ) e. Y )
97	95, 86	ffvelrnd 6038	. . . . . . . . 9 |- ( ( ( ( ( ph /\ F : X --> Y ) /\ d e. RR+ ) /\ c e. RR+ ) /\ ( x e. X /\ y e. X ) ) -> ( F ` x ) e. Y )
98		elbl4 21509	. . . . . . . . . 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 ) ) )
99		rpxr 11309	. . . . . . . . . . 11 |- ( d e. RR+ -> d e. RR* )
100		elbl3ps 21337	. . . . . . . . . . 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 ) )
101	99, 100	sylanl2 655	. . . . . . . . . 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 ) )
102	98, 101	bitr3d 258	. . . . . . . . 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 ) )
103	93, 94, 96, 97, 102	syl22anc 1265	. . . . . . . 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 ) )
104	92, 103	imbi12d 321	. . . . . . 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 ) ) )
105	104	2ralbidva 2874	. . . . . 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 ) ) )
106	105	rexbidva 2943	. . . . 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 ) ) )
107	106	ralbidva 2868	. . . 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 ) ) )
108	82, 107	bitrd 256	. . 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 ) ) )
109	108	pm5.32da 645	. 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 ) ) ) )
110	42, 109	bitrd 256	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 187   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   A.wral 2782   E.wrex 2783   (/)c0 3767   class class class wbr 4426   |-> cmpt 4484   X. cxp 4852   `'ccnv 4853   ran crn 4855   "cima 4857   -->wf 5597   ` cfv 5601  (class class class)co 6305   0cc0 9538   RR*cxr 9673   < clt 9674   RR+crp 11302   [,)cico 11637  PsMetcpsmet 18889   ballcbl 18892   fBascfbas 18893   filGencfg 18894  metUnifcmetu 18896  UnifOncust 21145   Cn_ucucn 21221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-psmet 18897  df-bl 18900  df-fbas 18902  df-fg 18903  df-metu 18904  df-fil 20792  df-ust 21146  df-ucn 21222

This theorem is referenced by:  qqhucn  28635  heicant  31682