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Theorem funtransport 25869
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 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 ) ) ) )  <->  ( 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 6335 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  ->  ( 1st `  p )  e.  ( EE `  n
) )
8 xp1st 6335 . . . . . . . . . 10  |-  ( p  e.  ( ( EE
`  m )  X.  ( EE `  m
) )  ->  ( 1st `  p )  e.  ( EE `  m
) )
9 axdimuniq 25756 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  NN  /\  ( 1st `  p
)  e.  ( EE
`  n ) )  /\  ( m  e.  NN  /\  ( 1st `  p )  e.  ( EE `  m ) ) )  ->  n  =  m )
10 fveq2 5687 . . . . . . . . . . . . . . . . 17  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1110riotaeqdv 6509 . . . . . . . . . . . . . . . 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 2415 . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . 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 2795 . . . . . 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 2410 . . . . . . . 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 2687 . . . . . 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 4862 . . . . . . . . . 10  |-  ( n  =  m  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  m )  X.  ( EE `  m
) ) )
2928eleq2d 2471 . . . . . . . . 9  |-  ( n  =  m  ->  (
p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  p  e.  ( ( EE `  m )  X.  ( EE `  m ) ) ) )
3028eleq2d 2471 . . . . . . . . 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 2893 . . . . . 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 2287 . . . 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 6129 . 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 25868 . . 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 5433 . 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 1721   E*wmo 2255    =/= wne 2567   E.wrex 2667   <.cop 3777   class class class wbr 4172    X. cxp 4835   Fun wfun 5407   ` cfv 5413   {coprab 6041   1stc1st 6306   2ndc2nd 6307   iota_crio 6501   NNcn 9956   EEcee 25731    Btwn cbtwn 25732  Cgrccgr 25733  TransportToctransport 25867
This theorem is referenced by:  fvtransport  25870
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-z 10239  df-uz 10445  df-fz 11000  df-ee 25734  df-transport 25868
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