Theorem frgrancvvdeqlem4 30797

Description: Lemma 4 for frgrancvvdeq 30806. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)

Hypotheses

Ref	Expression
frgrancvvdeq.nx	|- D = ( <. V , E >. Neighbors X )
frgrancvvdeq.ny	|- N = ( <. V , E >. Neighbors Y )
frgrancvvdeq.x	|- ( ph -> X e. V )
frgrancvvdeq.y	|- ( ph -> Y e. V )
frgrancvvdeq.ne	|- ( ph -> X =/= Y )
frgrancvvdeq.xy	|- ( ph -> Y e/ D )
frgrancvvdeq.f	|- ( ph -> V FriendGrph E )
frgrancvvdeq.a	|- A = ( x e. D |-> ( iota_ y e. N { x , y } e. ran E ) )

Assertion

Ref	Expression
frgrancvvdeqlem4	|- ( ( ph /\ x e. D ) -> { ( iota_ y e. N { x , y } e. ran E ) } = ( ( <. V , E >. Neighbors x ) i^i N ) )

Distinct variable groups:   y, D   x, y, V   x, E, y   y, Y   ph, y   y, N

Allowed substitution hints:   ph( x)   A( x, y)   D( x)   N( x)   X( x, y)   Y( x)

Proof of Theorem frgrancvvdeqlem4

Dummy variables a b n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrancvvdeq.ny	. . 3 |- N = ( <. V , E >. Neighbors Y )
2	1	ineq2i 3660	. 2 |- ( ( <. V , E >. Neighbors x ) i^i N ) = ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) )
3		frgrancvvdeq.nx	. . . . . . 7 |- D = ( <. V , E >. Neighbors X )
4	3	eleq2i 2532	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frgrancvvdeq.f	. . . . . . . 8 |- ( ph -> V FriendGrph E )
6		frisusgra 30755	. . . . . . . 8 |- ( V FriendGrph E -> V USGrph E )
7	5, 6	syl 16	. . . . . . 7 |- ( ph -> V USGrph E )
8		nbgraisvtx 23521	. . . . . . 7 |- ( V USGrph E -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
9	7, 8	syl 16	. . . . . 6 |- ( ph -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
10	4, 9	syl5bi 217	. . . . 5 |- ( ph -> ( x e. D -> x e. V ) )
11	10	imp 429	. . . 4 |- ( ( ph /\ x e. D ) -> x e. V )
12		frgrancvvdeq.x	. . . . 5 |- ( ph -> X e. V )
13		frgrancvvdeq.y	. . . . 5 |- ( ph -> Y e. V )
14		frgrancvvdeq.ne	. . . . 5 |- ( ph -> X =/= Y )
15		frgrancvvdeq.xy	. . . . 5 |- ( ph -> Y e/ D )
16		frgrancvvdeq.a	. . . . 5 |- A = ( x e. D |-> ( iota_ y e. N { x , y } e. ran E ) )
17	3, 1, 12, 13, 14, 15, 5, 16	frgrancvvdeqlem1 30794	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18	5	adantr 465	. . . . 5 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
19		frisusgranb 30760	. . . . 5 |- ( V FriendGrph E -> A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } )
20	18, 19	syl 16	. . . 4 |- ( ( ph /\ x e. D ) -> A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } )
21	11, 17, 20	jca31 534	. . 3 |- ( ( ph /\ x e. D ) -> ( ( x e. V /\ Y e. ( V \ { x } ) ) /\ A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) )
22		sneq 3998	. . . . . . . . 9 |- ( a = x -> { a } = { x } )
23	22	difeq2d 3585	. . . . . . . 8 |- ( a = x -> ( V \ { a } ) = ( V \ { x } ) )
24		oveq2 6211	. . . . . . . . . . 11 |- ( a = x -> ( <. V , E >. Neighbors a ) = ( <. V , E >. Neighbors x ) )
25	24	ineq1d 3662	. . . . . . . . . 10 |- ( a = x -> ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) )
26	25	eqeq1d 2456	. . . . . . . . 9 |- ( a = x -> ( ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } <-> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) )
27	26	rexbidv 2868	. . . . . . . 8 |- ( a = x -> ( E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } <-> E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) )
28	23, 27	raleqbidv 3037	. . . . . . 7 |- ( a = x -> ( A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } <-> A. b e. ( V \ { x } ) E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) )
29	28	rspcva 3177	. . . . . 6 |- ( ( x e. V /\ A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) -> A. b e. ( V \ { x } ) E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } )
30		oveq2 6211	. . . . . . . . . . . 12 |- ( b = Y -> ( <. V , E >. Neighbors b ) = ( <. V , E >. Neighbors Y ) )
31	30	ineq2d 3663	. . . . . . . . . . 11 |- ( b = Y -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) )
32	31	eqeq1d 2456	. . . . . . . . . 10 |- ( b = Y -> ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } <-> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } ) )
33	32	rexbidv 2868	. . . . . . . . 9 |- ( b = Y -> ( E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } <-> E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } ) )
34	33	rspcva 3177	. . . . . . . 8 |- ( ( Y e. ( V \ { x } ) /\ A. b e. ( V \ { x } ) E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) -> E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } )
35		ssnid 4017	. . . . . . . . . . . . . . 15 |- n e. { n }
36		eleq2 2527	. . . . . . . . . . . . . . . . 17 |- ( { n } = ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) -> ( n e. { n } <-> n e. ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) ) )
37	36	eqcoms 2466	. . . . . . . . . . . . . . . 16 |- ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( n e. { n } <-> n e. ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) ) )
38		elin 3650	. . . . . . . . . . . . . . . . 17 |- ( n e. ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) <-> ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) )
39	38	biimpi 194	. . . . . . . . . . . . . . . 16 |- ( n e. ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) -> ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) )
40	37, 39	syl6bi 228	. . . . . . . . . . . . . . 15 |- ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( n e. { n } -> ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) ) )
41	35, 40	mpi 17	. . . . . . . . . . . . . 14 |- ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) )
42		nbgraeledg 23520	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { n , x } e. ran E ) )
43		prcom 4064	. . . . . . . . . . . . . . . . . . . . . . 23 |- { n , x } = { x , n }
44	43	eleq1i 2531	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { n , x } e. ran E <-> { x , n } e. ran E )
45	42, 44	syl6bb 261	. . . . . . . . . . . . . . . . . . . . 21 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { x , n } e. ran E ) )
46	45	biimpd 207	. . . . . . . . . . . . . . . . . . . 20 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
47	7, 46	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
48	47	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. D ) -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
49	48	com12 31	. . . . . . . . . . . . . . . . 17 |- ( n e. ( <. V , E >. Neighbors x ) -> ( ( ph /\ x e. D ) -> { x , n } e. ran E ) )
50	49	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) -> ( ( ph /\ x e. D ) -> { x , n } e. ran E ) )
51	50	imp 429	. . . . . . . . . . . . . . 15 |- ( ( ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) /\ ( ph /\ x e. D ) ) -> { x , n } e. ran E )
52	1	eqcomi 2467	. . . . . . . . . . . . . . . . . . 19 |- ( <. V , E >. Neighbors Y ) = N
53	52	eleq2i 2532	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( <. V , E >. Neighbors Y ) <-> n e. N )
54	53	biimpi 194	. . . . . . . . . . . . . . . . 17 |- ( n e. ( <. V , E >. Neighbors Y ) -> n e. N )
55	54	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) -> n e. N )
56	3, 1, 12, 13, 14, 15, 5, 16	frgrancvvdeqlem3 30796	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )
57		preq2 4066	. . . . . . . . . . . . . . . . . 18 |- ( y = n -> { x , y } = { x , n } )
58	57	eleq1d 2523	. . . . . . . . . . . . . . . . 17 |- ( y = n -> ( { x , y } e. ran E <-> { x , n } e. ran E ) )
59	58	riota2 6187	. . . . . . . . . . . . . . . 16 |- ( ( n e. N /\ E! y e. N { x , y } e. ran E ) -> ( { x , n } e. ran E <-> ( iota_ y e. N { x , y } e. ran E ) = n ) )
60	55, 56, 59	syl2an 477	. . . . . . . . . . . . . . 15 |- ( ( ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) /\ ( ph /\ x e. D ) ) -> ( { x , n } e. ran E <-> ( iota_ y e. N { x , y } e. ran E ) = n ) )
61	51, 60	mpbid 210	. . . . . . . . . . . . . 14 |- ( ( ( n e. ( <. V , E >. Neighbors x ) /\ n e. ( <. V , E >. Neighbors Y ) ) /\ ( ph /\ x e. D ) ) -> ( iota_ y e. N { x , y } e. ran E ) = n )
62	41, 61	sylan 471	. . . . . . . . . . . . 13 |- ( ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( iota_ y e. N { x , y } e. ran E ) = n )
63	62	eqcomd 2462	. . . . . . . . . . . 12 |- ( ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> n = ( iota_ y e. N { x , y } e. ran E ) )
64	63	sneqd 4000	. . . . . . . . . . 11 |- ( ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> { n } = { ( iota_ y e. N { x , y } e. ran E ) } )
65		eqeq1 2458	. . . . . . . . . . . 12 |- ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } <-> { n } = { ( iota_ y e. N { x , y } e. ran E ) } ) )
66	65	adantr 465	. . . . . . . . . . 11 |- ( ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } <-> { n } = { ( iota_ y e. N { x , y } e. ran E ) } ) )
67	64, 66	mpbird 232	. . . . . . . . . 10 |- ( ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } )
68	67	ex 434	. . . . . . . . 9 |- ( ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) )
69	68	rexlimivw 2943	. . . . . . . 8 |- ( E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { n } -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) )
70	34, 69	syl 16	. . . . . . 7 |- ( ( Y e. ( V \ { x } ) /\ A. b e. ( V \ { x } ) E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) )
71	70	expcom 435	. . . . . 6 |- ( A. b e. ( V \ { x } ) E. n e. V ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors b ) ) = { n } -> ( Y e. ( V \ { x } ) -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) ) )
72	29, 71	syl 16	. . . . 5 |- ( ( x e. V /\ A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) -> ( Y e. ( V \ { x } ) -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) ) )
73	72	impancom 440	. . . 4 |- ( ( x e. V /\ Y e. ( V \ { x } ) ) -> ( A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) ) )
74	73	imp 429	. . 3 |- ( ( ( x e. V /\ Y e. ( V \ { x } ) ) /\ A. a e. V A. b e. ( V \ { a } ) E. n e. V ( ( <. V , E >. Neighbors a ) i^i ( <. V , E >. Neighbors b ) ) = { n } ) -> ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } ) )
75	21, 74	mpcom 36	. 2 |- ( ( ph /\ x e. D ) -> ( ( <. V , E >. Neighbors x ) i^i ( <. V , E >. Neighbors Y ) ) = { ( iota_ y e. N { x , y } e. ran E ) } )
76	2, 75	syl5req 2508	1 |- ( ( ph /\ x e. D ) -> { ( iota_ y e. N { x , y } e. ran E ) } = ( ( <. V , E >. Neighbors x ) i^i N ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   e/ wnel 2649   A.wral 2799   E.wrex 2800   E!wreu 2801   \ cdif 3436   i^i cin 3438   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403   |-> cmpt 4461   ran crn 4952   iota_crio 6163  (class class class)co 6203   USGrph cusg 23443   Neighbors cnbgra 23508   FriendGrph cfrgra 30751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-hash 12225  df-usgra 23445  df-nbgra 23511  df-frgra 30752

This theorem is referenced by:  frgrancvvdeqlem6  30799