Theorem frgrancvvdeqlem4 24857

Description: Lemma 4 for frgrancvvdeq 24866. 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 3702	. 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 2545	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frgrancvvdeq.f	. . . . . . . 8 |- ( ph -> V FriendGrph E )
6		frisusgra 24815	. . . . . . . 8 |- ( V FriendGrph E -> V USGrph E )
7	5, 6	syl 16	. . . . . . 7 |- ( ph -> V USGrph E )
8		nbgraisvtx 24254	. . . . . . 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 24854	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18	5	adantr 465	. . . . 5 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
19		frisusgranb 24820	. . . . 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 4043	. . . . . . . . 9 |- ( a = x -> { a } = { x } )
23	22	difeq2d 3627	. . . . . . . 8 |- ( a = x -> ( V \ { a } ) = ( V \ { x } ) )
24		oveq2 6303	. . . . . . . . . . 11 |- ( a = x -> ( <. V , E >. Neighbors a ) = ( <. V , E >. Neighbors x ) )
25	24	ineq1d 3704	. . . . . . . . . 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 2469	. . . . . . . . 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 2978	. . . . . . . 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 3077	. . . . . . 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 3217	. . . . . 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 6303	. . . . . . . . . . . 12 |- ( b = Y -> ( <. V , E >. Neighbors b ) = ( <. V , E >. Neighbors Y ) )
31	30	ineq2d 3705	. . . . . . . . . . 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 2469	. . . . . . . . . 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 2978	. . . . . . . . 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 3217	. . . . . . . 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 4062	. . . . . . . . . . . . . . 15 |- n e. { n }
36		eleq2 2540	. . . . . . . . . . . . . . . . 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 2479	. . . . . . . . . . . . . . . 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 3692	. . . . . . . . . . . . . . . . 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 24253	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { n , x } e. ran E ) )
43		prcom 4111	. . . . . . . . . . . . . . . . . . . . . . 23 |- { n , x } = { x , n }
44	43	eleq1i 2544	. . . . . . . . . . . . . . . . . . . . . 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 2480	. . . . . . . . . . . . . . . . . . 19 |- ( <. V , E >. Neighbors Y ) = N
53	52	eleq2i 2545	. . . . . . . . . . . . . . . . . 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 24856	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )
57		preq2 4113	. . . . . . . . . . . . . . . . . 18 |- ( y = n -> { x , y } = { x , n } )
58	57	eleq1d 2536	. . . . . . . . . . . . . . . . 17 |- ( y = n -> ( { x , y } e. ran E <-> { x , n } e. ran E ) )
59	58	riota2 6279	. . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . 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 4045	. . . . . . . . . . 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 2471	. . . . . . . . . . . 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 2956	. . . . . . . 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 2521	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 1379   e. wcel 1767   =/= wne 2662   e/ wnel 2663   A.wral 2817   E.wrex 2818   E!wreu 2819   \ cdif 3478   i^i cin 3480   {csn 4033   {cpr 4035   <.cop 4039   class class class wbr 4453   |-> cmpt 4511   ran crn 5006   iota_crio 6255  (class class class)co 6295   USGrph cusg 24153   Neighbors cnbgra 24240   FriendGrph cfrgra 24811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-frgra 24812

This theorem is referenced by:  frgrancvvdeqlem6  24859