Theorem funtransport 32138

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.

Step	Hyp	Ref	Expression
1		reeanv 3107	. . . . . 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 1061	. . . . . . . . . . 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 1061	. . . . . . . . . . 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 ) ) )
4	2, 3	anim12i 590	. . . . . . . . . 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 ) ) ) )
5	4	anim1i 592	. . . . . . . . 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 ) ) ) ) )
6	5	an4s 869	. . . . . . . 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 7198	. . . . . . . . . 10 |- ( p e. ( ( EE ` n ) X. ( EE ` n ) ) -> ( 1st ` p ) e. ( EE ` n ) )
8		xp1st 7198	. . . . . . . . . 10 |- ( p e. ( ( EE ` m ) X. ( EE ` m ) ) -> ( 1st ` p ) e. ( EE ` m ) )
9		axdimuniq 25793	. . . . . . . . . . . . 13 |- ( ( ( n e. NN /\ ( 1st ` p ) e. ( EE ` n ) ) /\ ( m e. NN /\ ( 1st ` p ) e. ( EE ` m ) ) ) -> n = m )
10		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( n = m -> ( EE ` n ) = ( EE ` m ) )
11	10	riotaeqdv 6612	. . . . . . . . . . . . . . . 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 ) ) )
12	11	eqeq2d 2632	. . . . . . . . . . . . . . 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 ) ) ) )
13	12	anbi2d 740	. . . . . . . . . . . . . 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 2643	. . . . . . . . . . . . . 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 )
15	13, 14	syl6bir 244	. . . . . . . . . . . . 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 ) )
16	9, 15	syl 17	. . . . . . . . . . . 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 ) )
17	16	an4s 869	. . . . . . . . . . 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 ) )
18	17	ex 450	. . . . . . . . . 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 ) ) )
19	7, 8, 18	syl2ani 688	. . . . . . . . 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 ) ) )
20	19	impd 447	. . . . . . . 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 ) )
21	6, 20	syl5 34	. . . . . . 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 ) )
22	21	rexlimivv 3036	. . . . . 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 )
23	1, 22	sylbir 225	. . . . 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 )
24	23	gen2 1723	. . . 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 2626	. . . . . . . 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 ) ) ) )
26	25	anbi2d 740	. . . . . . 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 ) ) ) ) )
27	26	rexbidv 3052	. . . . . 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 ) ) ) ) )
28	10	sqxpeqd 5141	. . . . . . . . . 10 |- ( n = m -> ( ( EE ` n ) X. ( EE ` n ) ) = ( ( EE ` m ) X. ( EE ` m ) ) )
29	28	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( p e. ( ( EE ` n ) X. ( EE ` n ) ) <-> p e. ( ( EE ` m ) X. ( EE ` m ) ) ) )
30	28	eleq2d 2687	. . . . . . . . 9 |- ( n = m -> ( q e. ( ( EE ` n ) X. ( EE ` n ) ) <-> q e. ( ( EE ` m ) X. ( EE ` m ) ) ) )
31	29, 30	3anbi12d 1400	. . . . . . . 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 ) ) ) )
32	31, 12	anbi12d 747	. . . . . . 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 ) ) ) ) )
33	32	cbvrexv 3172	. . . . . 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 ) ) ) )
34	27, 33	syl6bb 276	. . . . 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 ) ) ) ) )
35	34	mo4 2517	. . . 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 ) )
36	24, 35	mpbir 221	. . 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 ) ) )
37	36	funoprab 6760	. 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 32137	. . 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 ) ) ) }
39	38	funeqi 5909	. 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 ) ) ) } )
40	37, 39	mpbir 221	1 |- Fun TransportTo

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fun wfun 5882   ` cfv 5888   iota_crio 6610   {coprab 6651   1stc1st 7166   2ndc2nd 7167   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770  TransportToctransport 32136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771  df-transport 32137

This theorem is referenced by:  fvtransport  32139