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Theorem pstmfval 27741
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 27740 . . . 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 1016 . . 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 6294 . 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 5862 . . . . . 6  |-  (~Met `  D )  e.  _V
61, 5eqeltri 2525 . . . . 5  |-  .~  e.  _V
76ecelqsi 7365 . . . 4  |-  ( A  e.  X  ->  [ A ]  .~  e.  ( X /.  .~  ) )
873ad2ant2 1017 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  [ A ]  .~  e.  ( X /.  .~  ) )
96ecelqsi 7365 . . . 4  |-  ( B  e.  X  ->  [ B ]  .~  e.  ( X /.  .~  ) )
1093ad2ant3 1018 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  [ B ]  .~  e.  ( X /.  .~  ) )
11 rexeq 3039 . . . . . 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 2577 . . . . 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 4240 . . . 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 3039 . . . . . . 7  |-  ( y  =  [ B ]  .~  ->  ( E. b  e.  y  z  =  ( a D b )  <->  E. b  e.  [  B ]  .~  z  =  ( a D b ) ) )
1514rexbidv 2952 . . . . . 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 2577 . . . . 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 4240 . . . 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 7313 . . . . . . 7  |-  (  .~  e.  _V  ->  [ A ]  .~  e.  _V )
206, 19ax-mp 5 . . . . . 6  |-  [ A ]  .~  e.  _V
21 ecexg 7313 . . . . . . 7  |-  (  .~  e.  _V  ->  [ B ]  .~  e.  _V )
226, 21ax-mp 5 . . . . . 6  |-  [ B ]  .~  e.  _V
2320, 22ab2rexex 6772 . . . . 5  |-  { z  |  E. a  e. 
[  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  e.  _V
2423uniex 6577 . . . 4  |-  U. {
z  |  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) }  e.  _V
2513, 17, 18, 24ovmpt2 6419 . . 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 661 . 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 1003 . . . . . . . . . . 11  |-  ( ( ( 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 998 . . . . . . . . . . . 12  |-  ( ( ( 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 1001 . . . . . . . . . . . . . 14  |-  ( ( ( 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 27736 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
311, 30syl5eqss 3530 . . . . . . . . . . . . . . . . . . 19  |-  ( D  e.  (PsMet `  X
)  ->  .~  C_  ( X  X.  X ) )
32 xpss 5095 . . . . . . . . . . . . . . . . . . 19  |-  ( X  X.  X )  C_  ( _V  X.  _V )
3331, 32syl6ss 3498 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  (PsMet `  X
)  ->  .~  C_  ( _V  X.  _V ) )
34 df-rel 4992 . . . . . . . . . . . . . . . . . 18  |-  ( Rel 
.~ 
<->  .~  C_  ( _V  X.  _V ) )
3533, 34sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  (PsMet `  X
)  ->  Rel  .~  )
36353ad2ant1 1016 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  Rel  .~  )
3736adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  Rel  .~  )
38 relelec 7350 . . . . . . . . . . . . . . 15  |-  ( Rel 
.~  ->  ( e  e. 
[ A ]  .~  <->  A  .~  e ) )
3937, 38syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( 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 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  A  .~  e )
411breqi 4439 . . . . . . . . . . . . 13  |-  ( A  .~  e  <->  A (~Met `  D ) e )
4240, 41sylib 196 . . . . . . . . . . . 12  |-  ( ( ( 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 1002 . . . . . . . . . . . . . 14  |-  ( ( ( 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 7350 . . . . . . . . . . . . . . 15  |-  ( Rel 
.~  ->  ( f  e. 
[ B ]  .~  <->  B  .~  f ) )
4537, 44syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( 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 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  (
e  e.  [ A ]  .~  /\  f  e. 
[ B ]  .~  /\  z  =  ( e D f ) ) )  ->  B  .~  f )
471breqi 4439 . . . . . . . . . . . . 13  |-  ( B  .~  f  <->  B (~Met `  D ) f )
4846, 47sylib 196 . . . . . . . . . . . 12  |-  ( ( ( 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 27738 . . . . . . . . . . . 12  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D )
e  /\  B (~Met `  D ) f ) )  ->  ( A D B )  =  ( e D f ) )
5028, 42, 48, 49syl12anc 1225 . . . . . . . . . . 11  |-  ( ( ( 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 2485 . . . . . . . . . 10  |-  ( ( ( 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 714 . . . . . . . . 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 ) )
53523anassrs 1217 . . . . . . . 8  |-  ( ( ( ( ( ( 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 6284 . . . . . . . . . . . 12  |-  ( a  =  e  ->  (
a D b )  =  ( e D b ) )
5554eqeq2d 2455 . . . . . . . . . . 11  |-  ( a  =  e  ->  (
z  =  ( a D b )  <->  z  =  ( e D b ) ) )
56 oveq2 6285 . . . . . . . . . . . 12  |-  ( b  =  f  ->  (
e D b )  =  ( e D f ) )
5756eqeq2d 2455 . . . . . . . . . . 11  |-  ( b  =  f  ->  (
z  =  ( e D b )  <->  z  =  ( e D f ) ) )
5855, 57cbvrex2v 3077 . . . . . . . . . 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 ) )
5958biimpi 194 . . . . . . . . 9  |-  ( 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 466 . . . . . . . 8  |-  ( ( ( 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 2985 . . . . . . 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 ) )
62 simpl1 998 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  D  e.  (PsMet `  X
) )
63 simpl2 999 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  A  e.  X )
64 psmet0 20678 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
6562, 63, 64syl2anc 661 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A D A )  =  0 )
66 relelec 7350 . . . . . . . . . . 11  |-  ( Rel 
.~  ->  ( A  e. 
[ A ]  .~  <->  A  .~  A ) )
6762, 35, 663syl 20 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  e.  [ A ]  .~  <->  A  .~  A ) )
681a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  .~  =  (~Met `  D
) )
6968breqd 4444 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  .~  A  <->  A (~Met `  D ) A ) )
70 metidv 27737 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  A  e.  X )
)  ->  ( A
(~Met `  D ) A 
<->  ( A D A )  =  0 ) )
7162, 63, 63, 70syl12anc 1225 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A (~Met `  D ) A  <->  ( A D A )  =  0 ) )
7267, 69, 713bitrd 279 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( A  e.  [ A ]  .~  <->  ( A D A )  =  0 ) )
7365, 72mpbird 232 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  A  e.  [ A ]  .~  )
74 simpl3 1000 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  B  e.  X )
75 psmet0 20678 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X )  ->  ( B D B )  =  0 )
7662, 74, 75syl2anc 661 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B D B )  =  0 )
77 relelec 7350 . . . . . . . . . . 11  |-  ( Rel 
.~  ->  ( B  e. 
[ B ]  .~  <->  B  .~  B ) )
7862, 35, 773syl 20 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  e.  [ B ]  .~  <->  B  .~  B ) )
7968breqd 4444 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  .~  B  <->  B (~Met `  D ) B ) )
80 metidv 27737 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  ( B  e.  X  /\  B  e.  X )
)  ->  ( B
(~Met `  D ) B 
<->  ( B D B )  =  0 ) )
8162, 74, 74, 80syl12anc 1225 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B (~Met `  D ) B  <->  ( B D B )  =  0 ) )
8278, 79, 813bitrd 279 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
( B  e.  [ B ]  .~  <->  ( B D B )  =  0 ) )
8376, 82mpbird 232 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  ->  B  e.  [ B ]  .~  )
84 simpr 461 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  /\  z  =  ( A D B ) )  -> 
z  =  ( A D B ) )
85 rspceov 6317 . . . . . . . 8  |-  ( ( A  e.  [ A ]  .~  /\  B  e. 
[ B ]  .~  /\  z  =  ( A D B ) )  ->  E. a  e.  [  A ]  .~  E. b  e.  [  B ]  .~  z  =  ( a D b ) )
8673, 83, 84, 85syl3anc 1227 . . . . . . 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 ) )
8761, 86impbida 830 . . . . . 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 ) ) )
8887abbidv 2577 . . . . 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 ) } )
89 df-sn 4011 . . . . 5  |-  { ( A D B ) }  =  { z  |  z  =  ( A D B ) }
9088, 89syl6eqr 2500 . . . 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 ) } )
9190unieqd 4240 . . 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 ) } )
92 ovex 6305 . . . 4  |-  ( A D B )  e. 
_V
9392unisn 4245 . . 3  |-  U. {
( A D B ) }  =  ( A D B )
9491, 93syl6eq 2498 . 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 ) )
954, 26, 943eqtrd 2486 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 972    = wceq 1381    e. wcel 1802   {cab 2426   E.wrex 2792   _Vcvv 3093    C_ wss 3458   {csn 4010   U.cuni 4230   class class class wbr 4433    X. cxp 4983   Rel wrel 4990   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   [cec 7307   /.cqs 7308   0cc0 9490  PsMetcpsmet 18270  ~Metcmetid 27731  pstoMetcpstm 27732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-er 7309  df-ec 7311  df-qs 7315  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-xadd 11323  df-psmet 18279  df-metid 27733  df-pstm 27734
This theorem is referenced by:  pstmxmet  27742
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