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Theorem 2spotmdisj 25270
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 383 . . . . 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 455 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  V  /\  y  e.  V )  ->  x  e.  V )
4 3xpexg 6576 . . . . . . . . . . . . . . . . 17  |-  ( V  e.  _V  ->  (
( V  X.  V
)  X.  V )  e.  _V )
5 rabexg 4587 . . . . . . . . . . . . . . . . 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 )
64, 5syl 16 . . . . . . . . . . . . . . . 16  |-  ( V  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
7 eqeq2 2469 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  x  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  x ) )
87anbi2d 701 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  x  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
98rabbidv 3098 . . . . . . . . . . . . . . . . 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 ) } )
10 usgreghash2spot.m . . . . . . . . . . . . . . . . 17  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
119, 10fvmptg 5929 . . . . . . . . . . . . . . . 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 ) } )
123, 6, 11syl2anr 476 . . . . . . . . . . . . . . 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 ) } )
1312eleq2d 2524 . . . . . . . . . . . . . 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 ) } ) )
14 rabid 3031 . . . . . . . . . . . . . . 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 ) ) )
15 el2xptp 6816 . . . . . . . . . . . . . . . . . . 19  |-  ( t  e.  ( ( V  X.  V )  X.  V )  <->  E. b  e.  V  E. c  e.  V  E. d  e.  V  t  =  <. b ,  c ,  d >. )
16 ot2ndg 6788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( b  e.  V  /\  c  e.  V  /\  d  e.  V )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  c )
17163expa 1194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  c )
1817eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  x  <->  c  =  x ) )
1917adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  c )
2019eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y  <->  c  =  y ) )
21 ax-7 1795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( c  =  x  ->  (
c  =  y  ->  x  =  y )
)
2221adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
c  =  y  ->  x  =  y )
)
2320, 22sylbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  (
( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y  ->  x  =  y )
)
2423con3d 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( b  e.  V  /\  c  e.  V )  /\  d  e.  V )  /\  c  =  x )  ->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
)  =  y ) )
2524ex 432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( b  e.  V  /\  c  e.  V
)  /\  d  e.  V )  ->  (
c  =  x  -> 
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) ) )
2625a1dd 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 ) ) ) )
2718, 26sylbid 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 ) ) ) )
2827com12 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 ) ) ) )
2928a1i 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 ) ) ) ) )
30 fveq2 5848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  <. b ,  c ,  d >.  ->  ( 1st `  t )  =  ( 1st `  <. b ,  c ,  d
>. ) )
3130fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  <. b ,  c ,  d >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. b ,  c ,  d >. )
) )
3231eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  t ) )  =  x  <->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  x ) )
3331eqeq1d 2456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( 2nd `  ( 1st `  t ) )  =  y  <->  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) )
3433notbid 292 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  <. b ,  c ,  d >.  ->  ( -.  ( 2nd `  ( 1st `  t ) )  =  y  <->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) )
3534imbi2d 314 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  <. b ,  c ,  d >.  ->  (
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y )  <->  ( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  <. b ,  c ,  d
>. ) )  =  y ) ) )
3635imbi2d 314 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
3736imbi2d 314 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
3829, 32, 373imtr4d 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 ) ) ) ) )
3938com12 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 ) ) ) ) )
4039adantl 464 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
4140com13 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 ) ) ) ) )
4241rexlimdva 2946 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
4342rexlimivv 2951 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
4415, 43sylbi 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 ) ) ) )
4544imp 427 . . . . . . . . . . . . . . . . 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 ) ) )
4645com12 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 ) ) )
4746adantl 464 . . . . . . . . . . . . . . 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 ) ) )
4814, 47syl5bi 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 ) ) )
4913, 48sylbid 215 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
t  e.  ( M `
 x )  -> 
( -.  x  =  y  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5049com23 78 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  ( -.  x  =  y  ->  ( t  e.  ( M `  x )  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y ) ) )
5150imp31 430 . . . . . . . . . . 11  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  ( 2nd `  ( 1st `  t
) )  =  y )
5251intnand 914 . . . . . . . . . 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 ) )
5352intnand 914 . . . . . . . . 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 ) ) )
54 simpr 459 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  V  /\  y  e.  V )  ->  y  e.  V )
55 rabexg 4587 . . . . . . . . . . . . . . . 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 )
564, 55syl 16 . . . . . . . . . . . . . . 15  |-  ( V  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  y ) }  e.  _V )
57 eqeq2 2469 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  y ) )
5857anbi2d 701 . . . . . . . . . . . . . . . . 17  |-  ( a  =  y  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  y ) ) )
5958rabbidv 3098 . . . . . . . . . . . . . . . 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 ) } )
6059, 10fvmptg 5929 . . . . . . . . . . . . . . 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 ) } )
6154, 56, 60syl2anr 476 . . . . . . . . . . . . . 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 ) } )
6261eleq2d 2524 . . . . . . . . . . . . 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 ) } ) )
63 rabid 3031 . . . . . . . . . . . . 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 ) ) )
6462, 63syl6bb 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 ) ) ) )
6564notbid 292 . . . . . . . . . . 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 ) ) ) )
6665adantr 463 . . . . . . . . . 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 ) ) ) )
6766adantr 463 . . . . . . . . 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 ) ) ) )
6853, 67mpbird 232 . . . . . . . 8  |-  ( ( ( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  /\  -.  x  =  y
)  /\  t  e.  ( M `  x ) )  ->  -.  t  e.  ( M `  y
) )
6968ralrimiva 2868 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  A. t  e.  ( M `  x )  -.  t  e.  ( M `  y ) )
70 disj 3855 . . . . . . 7  |-  ( ( ( M `  x
)  i^i  ( M `  y ) )  =  (/) 
<-> 
A. t  e.  ( M `  x )  -.  t  e.  ( M `  y ) )
7169, 70sylibr 212 . . . . . 6  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) )
7271olcd 391 . . . . 5  |-  ( ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  /\  -.  x  =  y )  ->  ( x  =  y  \/  ( ( M `
 x )  i^i  ( M `  y
) )  =  (/) ) )
7372expcom 433 . . . 4  |-  ( -.  x  =  y  -> 
( ( V  e. 
_V  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  =  y  \/  ( ( M `
 x )  i^i  ( M `  y
) )  =  (/) ) ) )
742, 73pm2.61i 164 . . 3  |-  ( ( V  e.  _V  /\  ( x  e.  V  /\  y  e.  V
) )  ->  (
x  =  y  \/  ( ( M `  x )  i^i  ( M `  y )
)  =  (/) ) )
7574ralrimivva 2875 . 2  |-  ( V  e.  _V  ->  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  (
( M `  x
)  i^i  ( M `  y ) )  =  (/) ) )
76 fveq2 5848 . . 3  |-  ( x  =  y  ->  ( M `  x )  =  ( M `  y ) )
7776disjor 4424 . 2  |-  (Disj  x  e.  V  ( M `  x )  <->  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  (
( M `  x
)  i^i  ( M `  y ) )  =  (/) ) )
7875, 77sylibr 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    i^i cin 3460   (/)c0 3783   <.cotp 4024  Disj wdisj 4410    |-> cmpt 4497    X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   2SPathOnOt c2spthot 25058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-ot 4025  df-uni 4236  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  usgreghash2spot  25271
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