Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2spotmdisj Structured version   Unicode version

Theorem 2spotmdisj 30684
Description: The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
Assertion
Ref Expression
2spotmdisj  |-  ( V  e.  _V  -> Disj  x  e.  V  ( M `  x ) )
Distinct variable groups:    t, E, x, a    V, a, t, x    E, a    t, M, x
Allowed substitution hint:    M( a)

Proof of Theorem 2spotmdisj
Dummy variables  y 
c  d  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 385 . . . . 5  |-  ( x  =  y  ->  (
x  =  y  \/  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) ) )
21a1d 25 . . . 4  |-  ( x  =  y  ->  (
( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
x  =  y  \/  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) ) ) )
3 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  V  /\  y  e.  V )  ->  x  e.  V )
4 xpexg 6526 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V  e.  _V  /\  V  e.  _V )  ->  ( V  X.  V
)  e.  _V )
54anidms 645 . . . . . . . . . . . . . . . . . 18  |-  ( V  e.  _V  ->  ( V  X.  V )  e. 
_V )
6 xpexg 6526 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  X.  V
)  e.  _V  /\  V  e.  _V )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
75, 6mpancom 669 . . . . . . . . . . . . . . . . 17  |-  ( V  e.  _V  ->  (
( V  X.  V
)  X.  V )  e.  _V )
8 rabexg 4461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( V  X.  V
)  X.  V )  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
97, 8syl 16 . . . . . . . . . . . . . . . 16  |-  ( V  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
10 eqeq2 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  x  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  x ) )
1110anbi2d 703 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  x  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
1211rabbidv 2983 . . . . . . . . . . . . . . . . 17  |-  ( a  =  x  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } )
13 usgreghash2spot.m . . . . . . . . . . . . . . . . 17  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
1412, 13fvmptg 5791 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  V  /\  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )  ->  ( M `  x )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } )
153, 9, 14syl2anr 478 . . . . . . . . . . . . . . 15  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  ( M `  x )  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) } )
1615eleq2d 2510 . . . . . . . . . . . . . 14  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  ( M `
 x )  <->  t  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } ) )
17 rabid 2916 . . . . . . . . . . . . . . 15  |-  ( t  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) }  <-> 
( t  e.  ( ( V  X.  V
)  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
18 el2xptp 30149 . . . . . . . . . . . . . . . . . . 19  |-  ( t  e.  ( ( V  X.  V )  X.  V )  <->  E. b  e.  V  E. c  e.  V  E. d  e.  V  t  =  <. b ,  c ,  d >. )
19 ot2ndg 6611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( b  e.  V  /\  c  e.  V  /\  d  e.  V )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  c )
20193expa 1187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  c )
2120eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  x  <->  c  =  x ) )
2220adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  c )
2322eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y  <->  c  =  y ) )
24 ax-7 1728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( c  =  x  ->  (
c  =  y  ->  x  =  y )
)
2524adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
c  =  y  ->  x  =  y )
)
2623, 25sylbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y  ->  x  =  y )
)
2726con3d 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) )
2827ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
c  =  x  -> 
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) ) )
2928a1dd 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
c  =  x  -> 
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) ) ) )
3021, 29sylbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  x  -> 
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) ) ) )
3130com12 31 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  x  -> 
( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) ) ) )
3231a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  x  -> 
( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) ) ) ) )
33 fveq2 5710 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  <. b ,  c ,  d >.  ->  ( 1st `  t )  =  ( 1st `  <. b ,  c ,  d
>. ) )
3433fveq2d 5714 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  <. b ,  c ,  d >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
) )
3534eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  t ) )  =  x  <->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  x ) )
3634eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  t ) )  =  y  <->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) )
3736notbid 294 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  <. b ,  c ,  d >.  ->  ( -.  ( 2nd `  ( 1st `  t ) )  =  y  <->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) )
3837imbi2d 316 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y )  <->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) ) )
3938imbi2d 316 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) )  <-> 
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) ) ) )
4039imbi2d 316 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) )  <->  ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V
)  ->  ( (
x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) ) ) ) )
4132, 35, 403imtr4d 268 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  t ) )  =  x  ->  (
( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) ) )
4241com12 31 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 2nd `  ( 1st `  t ) )  =  x  ->  ( t  =  <. b ,  c ,  d >.  ->  (
( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) ) )
4342adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  ->  ( t  =  <. b ,  c ,  d >.  ->  (
( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) ) )
4443com13 80 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
t  =  <. b ,  c ,  d
>.  ->  ( ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x )  -> 
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) ) )
4544rexlimdva 2860 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( b  e.  V  /\  c  e.  V )  ->  ( E. d  e.  V  t  =  <. b ,  c ,  d
>.  ->  ( ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x )  -> 
( ( x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) ) )
4645rexlimivv 2865 . . . . . . . . . . . . . . . . . . 19  |-  ( E. b  e.  V  E. c  e.  V  E. d  e.  V  t  =  <. b ,  c ,  d >.  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  ->  ( (
x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) ) )
4718, 46sylbi 195 . . . . . . . . . . . . . . . . . 18  |-  ( t  e.  ( ( V  X.  V )  X.  V )  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  ->  ( (
x  e.  V  /\  y  e.  V )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) ) )
4847imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( t  e.  ( ( V  X.  V )  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) )  ->  (
( x  e.  V  /\  y  e.  V
)  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) )
4948com12 31 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  V  /\  y  e.  V )  ->  ( ( t  e.  ( ( V  X.  V )  X.  V
)  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5049adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
( t  e.  ( ( V  X.  V
)  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5117, 50syl5bi 217 . . . . . . . . . . . . . 14  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t ) )  =  y ) ) )
5216, 51sylbid 215 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  ( M `
 x )  -> 
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5352com23 78 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  ( -.  x  =  y  ->  ( t  e.  ( M `  x )  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5453imp31 432 . . . . . . . . . . 11  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y )
5554intnand 907 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  (
t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) )
5655intnand 907 . . . . . . . . 9  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  (
t  e.  ( ( V  X.  V )  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
57 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  V  /\  y  e.  V )  ->  y  e.  V )
58 rabexg 4461 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  X.  V
)  X.  V )  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) }  e.  _V )
597, 58syl 16 . . . . . . . . . . . . . . 15  |-  ( V  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) }  e.  _V )
60 eqeq2 2452 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  y ) )
6160anbi2d 703 . . . . . . . . . . . . . . . . 17  |-  ( a  =  y  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) )
6261rabbidv 2983 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) } )
6362, 13fvmptg 5791 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  V  /\  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) }  e.  _V )  ->  ( M `  y )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) } )
6457, 59, 63syl2anr 478 . . . . . . . . . . . . . 14  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  ( M `  y )  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) } )
6564eleq2d 2510 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  ( M `
 y )  <->  t  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) } ) )
66 rabid 2916 . . . . . . . . . . . . 13  |-  ( t  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) }  <-> 
( t  e.  ( ( V  X.  V
)  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) )
6765, 66syl6bb 261 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  ( M `
 y )  <->  ( t  e.  ( ( V  X.  V )  X.  V
)  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) )
6867notbid 294 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  ( -.  t  e.  ( M `  y )  <->  -.  ( t  e.  ( ( V  X.  V
)  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) )
6968adantr 465 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  ( -.  t  e.  ( M `  y
)  <->  -.  ( t  e.  ( ( V  X.  V )  X.  V
)  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) ) )
7069adantr 465 . . . . . . . . 9  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  ( -.  t  e.  ( M `  y )  <->  -.  (
t  e.  ( ( V  X.  V )  X.  V )  /\  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) ) ) )
7156, 70mpbird 232 . . . . . . . 8  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  t  e.  ( M `  y
) )
7271ralrimiva 2818 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  A. t  e.  ( M `  x )  -.  t  e.  ( M `  y ) )
73 disj 3738 . . . . . . 7  |-  ( ( ( M `  x
)  i^i  ( M `  y ) )  =  (/) 
<-> 
A. t  e.  ( M `  x )  -.  t  e.  ( M `  y ) )
7472, 73sylibr 212 . . . . . 6  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) )
7574olcd 393 . . . . 5  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  ( x  =  y  \/  ( ( M `
 x )  i^i  ( M `  y
) )  =  (/) ) )
7675expcom 435 . . . 4  |-  ( -.  x  =  y  -> 
( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  =  y  \/  ( ( M `
 x )  i^i  ( M `  y
) )  =  (/) ) ) )
772, 76pm2.61i 164 . . 3  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
x  =  y  \/  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) ) )
7877ralrimivva 2827 . 2  |-  ( V  e.  _V  ->  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  (
( M `  x
)  i^i  ( M `  y ) )  =  (/) ) )
79 fveq2 5710 . . 3  |-  ( x  =  y  ->  ( M `  x )  =  ( M `  y ) )
8079disjor 4295 . 2  |-  (Disj  x  e.  V  ( M `  x )  <->  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  (
( M `  x
)  i^i  ( M `  y ) )  =  (/) ) )
8178, 80sylibr 212 1  |-  ( V  e.  _V  -> Disj  x  e.  V  ( M `  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734   E.wrex 2735   {crab 2738   _Vcvv 2991    i^i cin 3346   (/)c0 3656   <.cotp 3904  Disj wdisj 4281    e. cmpt 4369    X. cxp 4857   ` cfv 5437  (class class class)co 6110   1stc1st 6594   2ndc2nd 6595   2SPathOnOt c2spthot 30398
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-ot 3905  df-uni 4111  df-iun 4192  df-disj 4282  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fv 5445  df-1st 6596  df-2nd 6597
This theorem is referenced by:  usgreghash2spot  30685
  Copyright terms: Public domain W3C validator