Theorem frgrancvvdeqlem4 25759

Description: Lemma 4 for frgrancvvdeq 25768. 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 3661	. 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 2499	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frgrancvvdeq.f	. . . . . . . 8 |- ( ph -> V FriendGrph E )
6		frisusgra 25718	. . . . . . . 8 |- ( V FriendGrph E -> V USGrph E )
7	5, 6	syl 17	. . . . . . 7 |- ( ph -> V USGrph E )
8		nbgraisvtx 25157	. . . . . . 7 |- ( V USGrph E -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
9	7, 8	syl 17	. . . . . 6 |- ( ph -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
10	4, 9	syl5bi 220	. . . . 5 |- ( ph -> ( x e. D -> x e. V ) )
11	10	imp 430	. . . 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 25756	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18	5	adantr 466	. . . . 5 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
19		frisusgranb 25723	. . . . 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 17	. . . 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 536	. . 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 4008	. . . . . . . . 9 |- ( a = x -> { a } = { x } )
23	22	difeq2d 3583	. . . . . . . 8 |- ( a = x -> ( V \ { a } ) = ( V \ { x } ) )
24		oveq2 6313	. . . . . . . . . . 11 |- ( a = x -> ( <. V , E >. Neighbors a ) = ( <. V , E >. Neighbors x ) )
25	24	ineq1d 3663	. . . . . . . . . 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 2424	. . . . . . . . 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 2936	. . . . . . . 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 3036	. . . . . . 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 3180	. . . . . 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 6313	. . . . . . . . . . . 12 |- ( b = Y -> ( <. V , E >. Neighbors b ) = ( <. V , E >. Neighbors Y ) )
31	30	ineq2d 3664	. . . . . . . . . . 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 2424	. . . . . . . . . 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 2936	. . . . . . . . 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 3180	. . . . . . . 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 4027	. . . . . . . . . . . . . . 15 |- n e. { n }
36		eleq2 2496	. . . . . . . . . . . . . . . . 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 2434	. . . . . . . . . . . . . . . 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 3649	. . . . . . . . . . . . . . . . 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 197	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 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 20	. . . . . . . . . . . . . 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 25156	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { n , x } e. ran E ) )
43		prcom 4078	. . . . . . . . . . . . . . . . . . . . . . 23 |- { n , x } = { x , n }
44	43	eleq1i 2498	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { n , x } e. ran E <-> { x , n } e. ran E )
45	42, 44	syl6bb 264	. . . . . . . . . . . . . . . . . . . . 21 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { x , n } e. ran E ) )
46	45	biimpd 210	. . . . . . . . . . . . . . . . . . . 20 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
47	7, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
48	47	adantr 466	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. D ) -> ( n e. ( <. V , E >. Neighbors x ) -> { x , n } e. ran E ) )
49	48	com12 32	. . . . . . . . . . . . . . . . 17 |- ( n e. ( <. V , E >. Neighbors x ) -> ( ( ph /\ x e. D ) -> { x , n } e. ran E ) )
50	49	adantr 466	. . . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . . 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 2435	. . . . . . . . . . . . . . . . . . 19 |- ( <. V , E >. Neighbors Y ) = N
53	52	eleq2i 2499	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( <. V , E >. Neighbors Y ) <-> n e. N )
54	53	biimpi 197	. . . . . . . . . . . . . . . . 17 |- ( n e. ( <. V , E >. Neighbors Y ) -> n e. N )
55	54	adantl 467	. . . . . . . . . . . . . . . 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 25758	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )
57		preq2 4080	. . . . . . . . . . . . . . . . . 18 |- ( y = n -> { x , y } = { x , n } )
58	57	eleq1d 2491	. . . . . . . . . . . . . . . . 17 |- ( y = n -> ( { x , y } e. ran E <-> { x , n } e. ran E ) )
59	58	riota2 6289	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 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 213	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 2430	. . . . . . . . . . . 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 4010	. . . . . . . . . . 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 2426	. . . . . . . . . . . 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 466	. . . . . . . . . . 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 235	. . . . . . . . . 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 435	. . . . . . . . 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 2911	. . . . . . . 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 17	. . . . . . 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 436	. . . . . 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 17	. . . . 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 441	. . . 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 430	. . 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 37	. 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 2476	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 187   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2614   e/ wnel 2615   A.wral 2771   E.wrex 2772   E!wreu 2773   \ cdif 3433   i^i cin 3435   {csn 3998   {cpr 4000   <.cop 4004   class class class wbr 4423   |-> cmpt 4482   ran crn 4854   iota_crio 6266  (class class class)co 6305   USGrph cusg 25055   Neighbors cnbgra 25143   FriendGrph cfrgra 25714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522  df-usgra 25058  df-nbgra 25146  df-frgra 25715

This theorem is referenced by:  frgrancvvdeqlem6  25761