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Theorem pstmfval 26259
Description: Function value of the metric induced by a pseudometric  D (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1  |-  .~  =  (~Met `  D )
Assertion
Ref Expression
pstmfval  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( [ A ]  .~  (pstoMet `  D ) [ B ]  .~  )  =  ( A D B ) )

Proof of Theorem pstmfval
Dummy variables  a 
b  x  y  z  e  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pstmval.1 . . . . 5  |-  .~  =  (~Met `  D )
21pstmval 26258 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  =  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } ) )
323ad2ant1 1004 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (pstoMet `  D )  =  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) } ) )
43oveqd 6107 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( [ A ]  .~  (pstoMet `  D ) [ B ]  .~  )  =  ( [ A ]  .~  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) } ) [ B ]  .~  ) )
5 fvex 5698 . . . . . 6  |-  (~Met `  D )  e.  _V
61, 5eqeltri 2511 . . . . 5  |-  .~  e.  _V
76ecelqsi 7152 . . . 4  |-  ( A  e.  X  ->  [ A ]  .~  e.  ( X /.  .~  ) )
873ad2ant2 1005 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  [ A ]  .~  e.  ( X /.  .~  ) )
96ecelqsi 7152 . . . 4  |-  ( B  e.  X  ->  [ B ]  .~  e.  ( X /.  .~  ) )
1093ad2ant3 1006 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  [ B ]  .~  e.  ( X /.  .~  ) )
11 rexeq 2916 . . . . . 6  |-  ( x  =  [ A ]  .~  ->  ( E. a  e.  x  E. b  e.  y  z  =  ( a D b )  <->  E. a  e.  [  A ]  .~  E. b  e.  y  z  =  ( a D b ) ) )
1211abbidv 2555 . . . . 5  |-  ( x  =  [ A ]  .~  ->  { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) }  =  { z  |  E. a  e.  [  A ]  .~  E. b  e.  y  z  =  ( a D b ) } )
1312unieqd 4098 . . . 4  |-  ( x  =  [ A ]  .~  ->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) }  =  U. { z  |  E. a  e. 
[  A ]  .~  E. b  e.  y  z  =  ( a D b ) } )
14 rexeq 2916 . . . . . . 7  |-  ( y  =  [ B ]  .~  ->  ( E. b  e.  y  z  =  ( a D b )  <->  E. b  e.  [  B ]  .~  z  =  ( a D b ) ) )
1514rexbidv 2734 . . . . . 6  |-  ( y  =  [ B ]  .~  ->  ( E. a  e.  [  A ]  .~  E. b  e.  y  z  =  ( a D b )  <->  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) ) )
1615abbidv 2555 . . . . 5  |-  ( y  =  [ B ]  .~  ->  { z  |  E. a  e.  [  A ]  .~  E. b  e.  y  z  =  ( a D b ) }  =  {
z  |  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) } )
1716unieqd 4098 . . . 4  |-  ( y  =  [ B ]  .~  ->  U. { z  |  E. a  e.  [  A ]  .~  E. b  e.  y  z  =  ( a D b ) }  =  U. { z  |  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) } )
18 eqid 2441 . . . 4  |-  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )  =  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )
19 ecexg 7101 . . . . . . 7  |-  (  .~  e.  _V  ->  [ A ]  .~  e.  _V )
206, 19ax-mp 5 . . . . . 6  |-  [ A ]  .~  e.  _V
21 ecexg 7101 . . . . . . 7  |-  (  .~  e.  _V  ->  [ B ]  .~  e.  _V )
226, 21ax-mp 5 . . . . . 6  |-  [ B ]  .~  e.  _V
2320, 22ab2rexex 6567 . . . . 5  |-  { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  e.  _V
2423uniex 6375 . . . 4  |-  U. {
z  |  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  e.  _V
2513, 17, 18, 24ovmpt2 6225 . . 3  |-  ( ( [ A ]  .~  e.  ( X /.  .~  )  /\  [ B ]  .~  e.  ( X /.  .~  ) )  ->  ( [ A ]  .~  (
x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) } ) [ B ]  .~  )  =  U. { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) } )
268, 10, 25syl2anc 656 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( [ A ]  .~  (
x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) } ) [ B ]  .~  )  =  U. { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) } )
27 simpr3 991 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  z  =  ( e D f ) )
28 simpl1 986 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  D  e.  (PsMet `  X ) )
29 simpr1 989 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  e  e.  [ A ]  .~  )
30 metidss 26254 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
311, 30syl5eqss 3397 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  (PsMet `  X
)  ->  .~  C_  ( X  X.  X ) )
32 xpss 4942 . . . . . . . . . . . . . . . . . . . 20  |-  ( X  X.  X )  C_  ( _V  X.  _V )
3331, 32syl6ss 3365 . . . . . . . . . . . . . . . . . . 19  |-  ( D  e.  (PsMet `  X
)  ->  .~  C_  ( _V  X.  _V ) )
34 df-rel 4843 . . . . . . . . . . . . . . . . . . 19  |-  ( Rel 
.~ 
<->  .~  C_  ( _V  X.  _V ) )
3533, 34sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  (PsMet `  X
)  ->  Rel  .~  )
36353ad2ant1 1004 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  Rel  .~  )
3736adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  Rel  .~  )
38 relelec 7137 . . . . . . . . . . . . . . . 16  |-  ( Rel 
.~  ->  ( e  e. 
[ A ]  .~  <->  A  .~  e ) )
3937, 38syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  ( e  e.  [ A ]  .~  <->  A  .~  e ) )
4029, 39mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  A  .~  e )
411breqi 4295 . . . . . . . . . . . . . 14  |-  ( A  .~  e  <->  A (~Met `  D ) e )
4240, 41sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  A (~Met `  D ) e )
43 simpr2 990 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  f  e.  [ B ]  .~  )
44 relelec 7137 . . . . . . . . . . . . . . . 16  |-  ( Rel 
.~  ->  ( f  e. 
[ B ]  .~  <->  B  .~  f ) )
4537, 44syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  ( f  e.  [ B ]  .~  <->  B  .~  f ) )
4643, 45mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  B  .~  f )
471breqi 4295 . . . . . . . . . . . . . 14  |-  ( B  .~  f  <->  B (~Met `  D ) f )
4846, 47sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  B (~Met `  D ) f )
49 metideq 26256 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D )
e  /\  B (~Met `  D ) f ) )  ->  ( A D B )  =  ( e D f ) )
5028, 42, 48, 49syl12anc 1211 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  ( A D B )  =  ( e D f ) )
5127, 50eqtr4d 2476 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  z  =  ( A D B ) )
5251adantlr 709 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X
)  /\  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) )  /\  ( e  e.  [ A ]  .~  /\  f  e.  [ B ]  .~  /\  z  =  ( e D f ) ) )  ->  z  =  ( A D B ) )
53523anassrs 1204 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) )  /\  e  e.  [ A ]  .~  )  /\  f  e.  [ B ]  .~  )  /\  z  =  ( e D f ) )  -> 
z  =  ( A D B ) )
54 oveq1 6097 . . . . . . . . . . . . 13  |-  ( a  =  e  ->  (
a D b )  =  ( e D b ) )
5554eqeq2d 2452 . . . . . . . . . . . 12  |-  ( a  =  e  ->  (
z  =  ( a D b )  <->  z  =  ( e D b ) ) )
56 oveq2 6098 . . . . . . . . . . . . 13  |-  ( b  =  f  ->  (
e D b )  =  ( e D f ) )
5756eqeq2d 2452 . . . . . . . . . . . 12  |-  ( b  =  f  ->  (
z  =  ( e D b )  <->  z  =  ( e D f ) ) )
5855, 57cbvrex2v 2954 . . . . . . . . . . 11  |-  ( E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b )  <->  E. e  e.  [  A ]  .~  E. f  e.  [  B ]  .~  z  =  ( e D f ) )
5958biimpi 194 . . . . . . . . . 10  |-  ( E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b )  ->  E. e  e.  [  A ]  .~  E. f  e. 
[  B ]  .~  z  =  ( e D f ) )
6059adantl 463 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) )  ->  E. e  e.  [  A ]  .~  E. f  e.  [  B ]  .~  z  =  ( e D f ) )
6153, 60r19.29_2a 2862 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) )  ->  z  =  ( A D B ) )
6261ex 434 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b )  -> 
z  =  ( A D B ) ) )
63 simpl1 986 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  D  e.  (PsMet `  X
) )
64 simpl2 987 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  A  e.  X )
65 psmet0 19843 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
6663, 64, 65syl2anc 656 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A D A )  =  0 )
67 relelec 7137 . . . . . . . . . . . 12  |-  ( Rel 
.~  ->  ( A  e. 
[ A ]  .~  <->  A  .~  A ) )
6863, 35, 673syl 20 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  e.  [ A ]  .~  <->  A  .~  A ) )
691a1i 11 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  .~  =  (~Met `  D
) )
7069breqd 4300 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  .~  A  <->  A (~Met `  D ) A ) )
71 metidv 26255 . . . . . . . . . . . 12  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  A  e.  X )
)  ->  ( A
(~Met `  D ) A 
<->  ( A D A )  =  0 ) )
7263, 64, 64, 71syl12anc 1211 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A (~Met `  D ) A  <->  ( A D A )  =  0 ) )
7368, 70, 723bitrd 279 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  e.  [ A ]  .~  <->  ( A D A )  =  0 ) )
7466, 73mpbird 232 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  A  e.  [ A ]  .~  )
75 simpl3 988 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  B  e.  X )
76 psmet0 19843 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X )  ->  ( B D B )  =  0 )
7763, 75, 76syl2anc 656 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B D B )  =  0 )
78 relelec 7137 . . . . . . . . . . . 12  |-  ( Rel 
.~  ->  ( B  e. 
[ B ]  .~  <->  B  .~  B ) )
7963, 35, 783syl 20 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  e.  [ B ]  .~  <->  B  .~  B ) )
8069breqd 4300 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  .~  B  <->  B (~Met `  D ) B ) )
81 metidv 26255 . . . . . . . . . . . 12  |-  ( ( D  e.  (PsMet `  X )  /\  ( B  e.  X  /\  B  e.  X )
)  ->  ( B
(~Met `  D ) B 
<->  ( B D B )  =  0 ) )
8263, 75, 75, 81syl12anc 1211 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B (~Met `  D ) B  <->  ( B D B )  =  0 ) )
8379, 80, 823bitrd 279 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  e.  [ B ]  .~  <->  ( B D B )  =  0 ) )
8477, 83mpbird 232 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  B  e.  [ B ]  .~  )
85 simpr 458 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
z  =  ( A D B ) )
86 oveq1 6097 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a D b )  =  ( A D b ) )
8786eqeq2d 2452 . . . . . . . . . 10  |-  ( a  =  A  ->  (
z  =  ( a D b )  <->  z  =  ( A D b ) ) )
88 oveq2 6098 . . . . . . . . . . 11  |-  ( b  =  B  ->  ( A D b )  =  ( A D B ) )
8988eqeq2d 2452 . . . . . . . . . 10  |-  ( b  =  B  ->  (
z  =  ( A D b )  <->  z  =  ( A D B ) ) )
9087, 89rspc2ev 3078 . . . . . . . . 9  |-  ( ( A  e.  [ A ]  .~  /\  B  e. 
[ B ]  .~  /\  z  =  ( A D B ) )  ->  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) )
9174, 84, 85, 90syl3anc 1213 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) )
9291ex 434 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
z  =  ( A D B )  ->  E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b ) ) )
9362, 92impbid 191 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( E. a  e.  [  A ]  .~  E. b  e. 
[  B ]  .~  z  =  ( a D b )  <->  z  =  ( A D B ) ) )
9493abbidv 2555 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  =  { z  |  z  =  ( A D B ) } )
95 df-sn 3875 . . . . 5  |-  { ( A D B ) }  =  { z  |  z  =  ( A D B ) }
9694, 95syl6eqr 2491 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  =  { ( A D B ) } )
9796unieqd 4098 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  U. {
z  |  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  =  U. {
( A D B ) } )
98 ovex 6115 . . . 4  |-  ( A D B )  e. 
_V
9998unisn 4103 . . 3  |-  U. {
( A D B ) }  =  ( A D B )
10097, 99syl6eq 2489 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  U. {
z  |  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  =  ( A D B ) )
1014, 26, 1003eqtrd 2477 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( [ A ]  .~  (pstoMet `  D ) [ B ]  .~  )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970    C_ wss 3325   {csn 3874   U.cuni 4088   class class class wbr 4289    X. cxp 4834   Rel wrel 4841   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   [cec 7095   /.cqs 7096   0cc0 9278  PsMetcpsmet 17759  ~Metcmetid 26249  pstoMetcpstm 26250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-xadd 11086  df-psmet 17768  df-metid 26251  df-pstm 26252
This theorem is referenced by:  pstmxmet  26260
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