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Theorem funtransport 25680
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 2819 . . . . . 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 6316 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  ->  ( 1st `  p )  e.  ( EE `  n
) )
8 xp1st 6316 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  ->  ( 1st `  p )  e.  ( EE `  m
) )
9 axdimuniq 25567 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  NN  /\  ( 1st `  p
)  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  ( 1st `  p )  e.  ( EE `  m ) ) )  ->  n  =  m )
10 fveq2 5669 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1110riotaeqdv 6487 . . . . . . . . . . . . . . . 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 2399 . . . . . . . . . . . . . . 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 2407 . . . . . . . . . . . . . 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 2779 . . . . . 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 1553 . . . 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 2394 . . . . . . . 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 2671 . . . . . 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 4844 . . . . . . . . . 10  |-  ( n  =  m  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  m )  X.  ( EE `  m
) ) )
2928eleq2d 2455 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3028eleq2d 2455 . . . . . . . . 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 2877 . . . . . 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 2272 . . . 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 6110 . 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 25679 . . 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 5415 . 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 1546    = wceq 1649    e. wcel 1717   E*wmo 2240    =/= wne 2551   E.wrex 2651   <.cop 3761   class class class wbr 4154    X. cxp 4817   Fun wfun 5389   ` cfv 5395   {coprab 6022   1stc1st 6287   2ndc2nd 6288   iota_crio 6479   NNcn 9933   EEcee 25542    Btwn cbtwn 25543  Cgrccgr 25544  TransportToctransport 25678
This theorem is referenced by:  fvtransport  25681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-z 10216  df-uz 10422  df-fz 10977  df-ee 25545  df-transport 25679
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