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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.
StepHypRef 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
) ) ) ) )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  (metUnif `  C )  =  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) ) ) )
51, 4syl5eq 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
) ) ) ) )
97, 8syl 16 . . . . . 6  |-  ( ph  ->  (metUnif `  D )  =  ( ( Y  X.  Y ) filGen ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) ) ) )
106, 9syl5eq 2486 . . . . 5  |-  ( ph  ->  V  =  ( ( Y  X.  Y )
filGen ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) ) ) )
115, 10oveq12d 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 ) ) ) ) ) )
1211eleq2d 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 ) )
1716imaeq2d 5168 . . . . . . . 8  |-  ( a  =  c  ->  ( `' C " ( 0 [,) a ) )  =  ( `' C " ( 0 [,) c
) ) )
1817cbvmptv 4382 . . . . . . 7  |-  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  =  ( c  e.  RR+  |->  ( `' C " ( 0 [,) c ) ) )
1918rneqi 5065 . . . . . 6  |-  ran  (
a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  =  ran  (
c  e.  RR+  |->  ( `' C " ( 0 [,) c ) ) )
2019metust 20142 . . . . 5  |-  ( ( X  =/=  (/)  /\  C  e.  (PsMet `  X )
)  ->  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) ) )  e.  (UnifOn `  X ) )
2115, 2, 20syl2anc 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 ) )
2423imaeq2d 5168 . . . . . . . 8  |-  ( b  =  d  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) d
) ) )
2524cbvmptv 4382 . . . . . . 7  |-  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
2625rneqi 5065 . . . . . 6  |-  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  =  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
2726metust 20142 . . . . 5  |-  ( ( Y  =/=  (/)  /\  D  e.  (PsMet `  Y )
)  ->  ( ( Y  X.  Y ) filGen ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) ) )  e.  (UnifOn `  Y ) )
2822, 7, 27syl2anc 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 ) )
3029imaeq2d 5168 . . . . . . . 8  |-  ( a  =  e  ->  ( `' C " ( 0 [,) a ) )  =  ( `' C " ( 0 [,) e
) ) )
3130cbvmptv 4382 . . . . . . 7  |-  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  =  ( e  e.  RR+  |->  ( `' C " ( 0 [,) e ) ) )
3231rneqi 5065 . . . . . 6  |-  ran  (
a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  =  ran  (
e  e.  RR+  |->  ( `' C " ( 0 [,) e ) ) )
3332metustfbas 20140 . . . . 5  |-  ( ( X  =/=  (/)  /\  C  e.  (PsMet `  X )
)  ->  ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
3415, 2, 33syl2anc 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 ) )
3635imaeq2d 5168 . . . . . . . 8  |-  ( b  =  f  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) f
) ) )
3736cbvmptv 4382 . . . . . . 7  |-  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  =  ( f  e.  RR+  |->  ( `' D " ( 0 [,) f ) ) )
3837rneqi 5065 . . . . . 6  |-  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  =  ran  (
f  e.  RR+  |->  ( `' D " ( 0 [,) f ) ) )
3938metustfbas 20140 . . . . 5  |-  ( ( Y  =/=  (/)  /\  D  e.  (PsMet `  Y )
)  ->  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  e.  ( fBas `  ( Y  X.  Y
) ) )
4022, 7, 39syl2anc 661 . . . 4  |-  ( ph  ->  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )  e.  ( fBas `  ( Y  X.  Y ) ) )
4113, 14, 21, 28, 34, 40isucn2 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 ) ) ) ) )
4212, 41bitrd 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 ) )
4544imaeq2d 5168 . . . . . . . . . . . 12  |-  ( f  =  d  ->  ( `' D " ( 0 [,) f ) )  =  ( `' D " ( 0 [,) d
) ) )
4645eqeq2d 2453 . . . . . . . . . . 11  |-  ( f  =  d  ->  (
( `' D "
( 0 [,) d
) )  =  ( `' D " ( 0 [,) f ) )  <-> 
( `' D "
( 0 [,) d
) )  =  ( `' D " ( 0 [,) d ) ) ) )
4746rspcev 3072 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) d
) ) )  ->  E. f  e.  RR+  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) f
) ) )
4843, 47mpan2 671 . . . . . . . . 9  |-  ( d  e.  RR+  ->  E. f  e.  RR+  ( `' D " ( 0 [,) d
) )  =  ( `' D " ( 0 [,) f ) ) )
4948adantl 466 . . . . . . . 8  |-  ( (
ph  /\  d  e.  RR+ )  ->  E. f  e.  RR+  ( `' D " ( 0 [,) d
) )  =  ( `' D " ( 0 [,) f ) ) )
5038metustel 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
) ) ) )
517, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( `' D " ( 0 [,) d
) )  e.  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  <->  E. f  e.  RR+  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) f
) ) ) )
5251adantr 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
) ) ) )
5349, 52mpbird 232 . . . . . . 7  |-  ( (
ph  /\  d  e.  RR+ )  ->  ( `' D " ( 0 [,) d ) )  e. 
ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) ) )
5426metustel 20126 . . . . . . . . 9  |-  ( D  e.  (PsMet `  Y
)  ->  ( v  e.  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )  <->  E. d  e.  RR+  v  =  ( `' D " ( 0 [,) d ) ) ) )
557, 54syl 16 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )  <->  E. d  e.  RR+  v  =  ( `' D " ( 0 [,) d ) ) ) )
5655biimpa 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 ) ) )
5857breqd 4302 . . . . . . . . . 10  |-  ( (
ph  /\  v  =  ( `' D " ( 0 [,) d ) ) )  ->  ( ( F `  x )
v ( F `  y )  <->  ( F `  x ) ( `' D " ( 0 [,) d ) ) ( F `  y
) ) )
5958imbi2d 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
) ) ) )
6059ralbidv 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
) ) ) )
6160rexralbidv 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
) ) ) )
6253, 56, 61ralxfrd 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 ) )
6564imaeq2d 5168 . . . . . . . . . . . . 13  |-  ( e  =  c  ->  ( `' C " ( 0 [,) e ) )  =  ( `' C " ( 0 [,) c
) ) )
6665eqeq2d 2453 . . . . . . . . . . . 12  |-  ( e  =  c  ->  (
( `' C "
( 0 [,) c
) )  =  ( `' C " ( 0 [,) e ) )  <-> 
( `' C "
( 0 [,) c
) )  =  ( `' C " ( 0 [,) c ) ) ) )
6766rspcev 3072 . . . . . . . . . . 11  |-  ( ( c  e.  RR+  /\  ( `' C " ( 0 [,) c ) )  =  ( `' C " ( 0 [,) c
) ) )  ->  E. e  e.  RR+  ( `' C " ( 0 [,) c ) )  =  ( `' C " ( 0 [,) e
) ) )
6863, 67mpan2 671 . . . . . . . . . 10  |-  ( c  e.  RR+  ->  E. e  e.  RR+  ( `' C " ( 0 [,) c
) )  =  ( `' C " ( 0 [,) e ) ) )
6968adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  RR+ )  ->  E. e  e.  RR+  ( `' C " ( 0 [,) c
) )  =  ( `' C " ( 0 [,) e ) ) )
7032metustel 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
) ) ) )
712, 70syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( `' C " ( 0 [,) c
) )  e.  ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  <->  E. e  e.  RR+  ( `' C " ( 0 [,) c ) )  =  ( `' C " ( 0 [,) e
) ) ) )
7271adantr 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
) ) ) )
7369, 72mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  c  e.  RR+ )  ->  ( `' C " ( 0 [,) c ) )  e. 
ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a
) ) ) )
7419metustel 20126 . . . . . . . . . 10  |-  ( C  e.  (PsMet `  X
)  ->  ( u  e.  ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a
) ) )  <->  E. c  e.  RR+  u  =  ( `' C " ( 0 [,) c ) ) ) )
752, 74syl 16 . . . . . . . . 9  |-  ( ph  ->  ( u  e.  ran  ( a  e.  RR+  |->  ( `' C " ( 0 [,) a ) ) )  <->  E. c  e.  RR+  u  =  ( `' C " ( 0 [,) c ) ) ) )
7675biimpa 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 ) ) )
7877breqd 4302 . . . . . . . . . 10  |-  ( (
ph  /\  u  =  ( `' C " ( 0 [,) c ) ) )  ->  ( x u y  <->  x ( `' C " ( 0 [,) c ) ) y ) )
7978imbi1d 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
) ) ) )
80792ralbidv 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
) ) ) )
8173, 76, 80rexxfrd 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 ) ) ) )
8281ralbidv 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
) ) ) )
8362, 82bitrd 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 ) ) ) )
8483adantr 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
) ) ) )
852ad4antr 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 ) )
9290, 91sylanl2 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 ) )
9389, 92bitr3d 255 . . . . . . . . 9  |-  ( ( ( C  e.  (PsMet `  X )  /\  c  e.  RR+ )  /\  (
y  e.  X  /\  x  e.  X )
)  ->  ( x
( `' C "
( 0 [,) c
) ) y  <->  ( x C y )  < 
c ) )
9485, 86, 87, 88, 93syl22anc 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
) )
957ad4antr 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 )
9897, 87ffvelrnd 5843 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  F : X --> Y )  /\  d  e.  RR+ )  /\  c  e.  RR+ )  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  y
)  e.  Y )
9997, 88ffvelrnd 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 ) )
103101, 102sylanl2 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 ) )
104100, 103bitr3d 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
) )
10595, 96, 98, 99, 104syl22anc 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
) )
10694, 105imbi12d 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 ) ) )
1071062ralbidva 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 ) ) )
108107rexbidva 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 ) ) )
109108ralbidva 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 ) ) )
11084, 109bitrd 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 ) ) )
111110pm5.32da 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 ) ) ) )
11242, 111bitrd 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 ) ) ) )
Colors of variables: wff setvar class
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   Cnucucn 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
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