Theorem wlkeq 16295

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 2231	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2231	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
3		eqid 2231	. . . . . . 7 |- ( 1st ` A ) = ( 1st ` A )
4		eqid 2231	. . . . . . 7 |- ( 2nd ` A ) = ( 2nd ` A )
5	1, 2, 3, 4	wlkelwrd 16294	. . . . . 6 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
6		eqid 2231	. . . . . . 7 |- ( 1st ` B ) = ( 1st ` B )
7		eqid 2231	. . . . . . 7 |- ( 2nd ` B ) = ( 2nd ` B )
8	1, 2, 6, 7	wlkelwrd 16294	. . . . . 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 16260	. . . . . . 7 |- ( A e. (Walks ` G ) -> G e. _V )
11		wlkcprim 16291	. . . . . . 7 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
12		wlklenvm1g 16283	. . . . . . 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 16260	. . . . . . 7 |- ( B e. (Walks ` G ) -> G e. _V )
15		wlkcprim 16291	. . . . . . 7 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
16		wlklenvm1g 16283	. . . . . . 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 11220	. . . . . . . 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 11183	. . . . . . . . 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 10438	. . . . . . . 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 1335	. . . . . . 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 1044	. . 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 2238	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( N = (♯ ` ( 1st ` B ) ) <-> (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) ) )
33		oveq2 6036	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
34	33	raleqdv 2737	. . . . . . 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 6036	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0 ... N ) = ( 0 ... (♯ ` ( 1st ` A ) ) ) )
37	36	raleqdv 2737	. . . . . . 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 1047	. . 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 16290	. . . 4 |- ( A e. (Walks ` G ) -> A e. ( _V X. _V ) )
44		wlkelvv 16290	. . . 4 |- ( B e. (Walks ` G ) -> B e. ( _V X. _V ) )
45		xpopth 6348	. . . 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 1044	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
48		3anass 1009	. . . 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 594	. . . 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 1005   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   class class class wbr 4093   X. cxp 4729   dom cdm 4731   -->wf 5329   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   0cc0 8092   1c1 8093   - cmin 8409   NN0cn0 9461   ...cfz 10305  ..^cfzo 10439  ♯chash 11100  Word cword 11179  Vtxcvtx 15953  iEdgciedg 15954  Walkscwlks 16258

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-ndx 13165  df-slot 13166  df-base 13168  df-edgf 15946  df-vtx 15955  df-iedg 15956  df-wlks 16259

This theorem is referenced by:  uspgr2wlkeq  16306