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Theorem funtransport 25965
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 2875 . . . . . 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 957 . . . . . . . . . . 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 957 . . . . . . . . . . 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 550 . . . . . . . . . 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 552 . . . . . . . . 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 800 . . . . . . . 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 6376 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  ->  ( 1st `  p )  e.  ( EE `  n
) )
8 xp1st 6376 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  ->  ( 1st `  p )  e.  ( EE `  m
) )
9 axdimuniq 25852 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  NN  /\  ( 1st `  p
)  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  ( 1st `  p )  e.  ( EE `  m ) ) )  ->  n  =  m )
10 fveq2 5728 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1110riotaeqdv 6550 . . . . . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . 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 2455 . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 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 16 . . . . . . . . . . . 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 800 . . . . . . . . . . 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 424 . . . . . . . . . 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 638 . . . . . . . . 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 421 . . . . . . . 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 30 . . . . . . 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 2835 . . . . . 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 205 . . . . 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 1556 . . . 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 2442 . . . . . . . 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 685 . . . . . . 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 2726 . . . . . 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 4903 . . . . . . . . . 10  |-  ( n  =  m  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  m )  X.  ( EE `  m
) ) )
2928eleq2d 2503 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3028eleq2d 2503 . . . . . . . . 9  |-  ( n  =  m  ->  (
q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  q  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3129, 303anbi12d 1255 . . . . . . . 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 692 . . . . . . 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 2933 . . . . . 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 253 . . . . 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 2314 . . . 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 201 . . 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 6170 . 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 25964 . . 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 5474 . 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 201 1  |-  Fun TransportTo
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2282    =/= wne 2599   E.wrex 2706   <.cop 3817   class class class wbr 4212    X. cxp 4876   Fun wfun 5448   ` cfv 5454   {coprab 6082   1stc1st 6347   2ndc2nd 6348   iota_crio 6542   NNcn 10000   EEcee 25827    Btwn cbtwn 25828  Cgrccgr 25829  TransportToctransport 25963
This theorem is referenced by:  fvtransport  25966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-z 10283  df-uz 10489  df-fz 11044  df-ee 25830  df-transport 25964
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