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Theorem funtransport 24726
Description: The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funtransport  |-  Fun TransportTo

Proof of Theorem funtransport
Dummy variables  m  n  p  q  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 2720 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  <->  ( E. n  e.  NN  ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  E. m  e.  NN  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
2 simp1 955 . . . . . . . . . . 11  |-  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  ->  p  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )
3 simp1 955 . . . . . . . . . . 11  |-  ( ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  ->  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) )
42, 3anim12i 549 . . . . . . . . . 10  |-  ( ( ( p  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  ( p  e.  (
( EE `  m
)  X.  ( EE
`  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) ) )  ->  ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  p  e.  ( ( EE `  m
)  X.  ( EE
`  m ) ) ) )
54anim1i 551 . . . . . . . . 9  |-  ( ( ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  ( p  e.  (
( EE `  m
)  X.  ( EE
`  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) ) )  /\  ( x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  ( (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  p  e.  ( ( EE `  m )  X.  ( EE `  m
) ) )  /\  ( x  =  ( iota_ r  e.  ( EE
`  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
65an4s 799 . . . . . . . 8  |-  ( ( ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  ( (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  p  e.  ( ( EE `  m )  X.  ( EE `  m
) ) )  /\  ( x  =  ( iota_ r  e.  ( EE
`  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
7 xp1st 6165 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  ->  ( 1st `  p )  e.  ( EE `  n
) )
8 xp1st 6165 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  ->  ( 1st `  p )  e.  ( EE `  m
) )
9 axdimuniq 24613 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  NN  /\  ( 1st `  p
)  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  ( 1st `  p )  e.  ( EE `  m ) ) )  ->  n  =  m )
10 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1110riotaeqdv 6321 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  ( iota_ r  e.  ( EE
`  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )
1211eqeq2d 2307 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  <->  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )
1312anbi2d 684 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  (
( x  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <-> 
( x  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
14 eqtr3 2315 . . . . . . . . . . . . . 14  |-  ( ( x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y )
1513, 14syl6bir 220 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  (
( x  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y ) )
169, 15syl 15 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  ( 1st `  p
)  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  ( 1st `  p )  e.  ( EE `  m ) ) )  ->  (
( x  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y ) )
1716an4s 799 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  m  e.  NN )  /\  ( ( 1st `  p )  e.  ( EE `  n )  /\  ( 1st `  p
)  e.  ( EE
`  m ) ) )  ->  ( (
x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y ) )
1817ex 423 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( ( 1st `  p )  e.  ( EE `  n )  /\  ( 1st `  p
)  e.  ( EE
`  m ) )  ->  ( ( x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) )  /\  y  =  (
iota_ r  e.  ( EE `  m ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y ) ) )
197, 8, 18syl2ani 637 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  p  e.  ( ( EE `  m
)  X.  ( EE
`  m ) ) )  ->  ( (
x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  ->  x  =  y ) ) )
2019imp3a 420 . . . . . . . 8  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) )  /\  ( x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) )  /\  y  =  (
iota_ r  e.  ( EE `  m ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  x  =  y ) )
216, 20syl5 28 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  /\  x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) ) )  /\  ( ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  /\  y  =  ( iota_ r  e.  ( EE `  m
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) ) ) )  ->  x  =  y ) )
2221rexlimivv 2685 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  x  =  y )
231, 22sylbir 204 . . . . 5  |-  ( ( E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  E. m  e.  NN  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  x  =  y )
2423gen2 1537 . . . 4  |-  A. x A. y ( ( E. n  e.  NN  (
( p  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  E. m  e.  NN  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  x  =  y )
25 eqeq1 2302 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  <->  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )
2625anbi2d 684 . . . . . . 7  |-  ( x  =  y  ->  (
( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <-> 
( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
2726rexbidv 2577 . . . . . 6  |-  ( x  =  y  ->  ( E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <->  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
2810, 10xpeq12d 4730 . . . . . . . . . 10  |-  ( n  =  m  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  m )  X.  ( EE `  m
) ) )
2928eleq2d 2363 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3028eleq2d 2363 . . . . . . . . 9  |-  ( n  =  m  ->  (
q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  q  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3129, 303anbi12d 1253 . . . . . . . 8  |-  ( n  =  m  ->  (
( p  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  <->  ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) ) ) )
3231, 12anbi12d 691 . . . . . . 7  |-  ( n  =  m  ->  (
( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <-> 
( ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
3332cbvrexv 2778 . . . . . 6  |-  ( E. n  e.  NN  (
( p  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <->  E. m  e.  NN  ( ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )
3427, 33syl6bb 252 . . . . 5  |-  ( x  =  y  ->  ( E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <->  E. m  e.  NN  ( ( p  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  q  e.  ( ( EE `  m
)  X.  ( EE
`  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) ) )
3534mo4 2189 . . . 4  |-  ( E* x E. n  e.  NN  ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <->  A. x A. y ( ( E. n  e.  NN  ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  /\  E. m  e.  NN  ( ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  /\  q  e.  ( ( EE `  m )  X.  ( EE `  m ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  y  =  ( iota_ r  e.  ( EE `  m ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) )  ->  x  =  y ) )
3624, 35mpbir 200 . . 3  |-  E* x E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )
3736funoprab 5960 . 2  |-  Fun  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) }
38 df-transport 24725 . . 3  |- TransportTo  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) }
3938funeqi 5291 . 2  |-  ( Fun TransportTo  <->  Fun  {
<. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) } )
4037, 39mpbir 200 1  |-  Fun TransportTo
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   E*wmo 2157    =/= wne 2459   E.wrex 2557   <.cop 3656   class class class wbr 4039    X. cxp 4703   Fun wfun 5265   ` cfv 5271   {coprab 5875   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   NNcn 9762   EEcee 24588    Btwn cbtwn 24589  Cgrccgr 24590  TransportToctransport 24724
This theorem is referenced by:  fvtransport  24727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-z 10041  df-uz 10247  df-fz 10799  df-ee 24591  df-transport 24725
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