Theorem wlkeq 16065

Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)

Assertion

Ref	Expression
wlkeq	|- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Distinct variable groups:   x, A   x, B   x, N

Allowed substitution hint:   G( x)

Proof of Theorem wlkeq

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2229	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
3		eqid 2229	. . . . . . 7 |- ( 1st ` A ) = ( 1st ` A )
4		eqid 2229	. . . . . . 7 |- ( 2nd ` A ) = ( 2nd ` A )
5	1, 2, 3, 4	wlkelwrd 16064	. . . . . 6 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
6		eqid 2229	. . . . . . 7 |- ( 1st ` B ) = ( 1st ` B )
7		eqid 2229	. . . . . . 7 |- ( 2nd ` B ) = ( 2nd ` B )
8	1, 2, 6, 7	wlkelwrd 16064	. . . . . 6 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
9	5, 8	anim12i 338	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) )
10		wlkmex 16032	. . . . . . 7 |- ( A e. (Walks ` G ) -> G e. _V )
11		wlkcprim 16061	. . . . . . 7 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
12		wlklenvm1g 16053	. . . . . . 7 |- ( ( G e. _V /\ ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
13	10, 11, 12	syl2anc 411	. . . . . 6 |- ( A e. (Walks ` G ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
14		wlkmex 16032	. . . . . . 7 |- ( B e. (Walks ` G ) -> G e. _V )
15		wlkcprim 16061	. . . . . . 7 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
16		wlklenvm1g 16053	. . . . . . 7 |- ( ( G e. _V /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
17	14, 15, 16	syl2anc 411	. . . . . 6 |- ( B e. (Walks ` G ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
18	13, 17	anim12i 338	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) )
19		eqwrd 11112	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 1st ` B ) e. Word dom (iEdg ` G ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
20	19	ad2ant2r 509	. . . . . . 7 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
21	20	adantr 276	. . . . . 6 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
22		lencl 11075	. . . . . . . . 9 |- ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
23	22	adantr 276	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
24		simpr 110	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) )
25		simpr 110	. . . . . . . 8 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) )
26		2ffzeq 10337	. . . . . . . 8 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
27	23, 24, 25, 26	syl2an3an 1332	. . . . . . 7 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
28	27	adantr 276	. . . . . 6 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
29	21, 28	anbi12d 473	. . . . 5 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
30	9, 18, 29	syl2anc 411	. . . 4 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
31	30	3adant3 1041	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
32		eqeq1 2236	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( N = (♯ ` ( 1st ` B ) ) <-> (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) ) )
33		oveq2 6009	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
34	33	raleqdv 2734	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) <-> A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) )
35	32, 34	anbi12d 473	. . . . . 6 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
36		oveq2 6009	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0 ... N ) = ( 0 ... (♯ ` ( 1st ` A ) ) ) )
37	36	raleqdv 2734	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) <-> A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
38	32, 37	anbi12d 473	. . . . . 6 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
39	35, 38	anbi12d 473	. . . . 5 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
40	39	bibi2d 232	. . . 4 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
41	40	3ad2ant3 1044	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
42	31, 41	mpbird 167	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
43		wlkelvv 16060	. . . 4 |- ( A e. (Walks ` G ) -> A e. ( _V X. _V ) )
44		wlkelvv 16060	. . . 4 |- ( B e. (Walks ` G ) -> B e. ( _V X. _V ) )
45		xpopth 6322	. . . 4 |- ( ( A e. ( _V X. _V ) /\ B e. ( _V X. _V ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
46	43, 44, 45	syl2an 289	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
47	46	3adant3 1041	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
48		3anass 1006	. . . 4 |- ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
49		anandi 592	. . . 4 |- ( ( N = (♯ ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
50	48, 49	bitr2i 185	. . 3 |- ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
51	50	a1i 9	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
52	42, 47, 51	3bitr3d 218	1 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   class class class wbr 4083   X. cxp 4717   dom cdm 4719   -->wf 5314   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   0cc0 7999   1c1 8000   - cmin 8317   NN0cn0 9369   ...cfz 10204  ..^cfzo 10338  ♯chash 10997  Word cword 11071  Vtxcvtx 15813  iEdgciedg 15814  Walkscwlks 16030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-wlks 16031

This theorem is referenced by:  uspgr2wlkeq  16076