Theorem frgrancvvdeqlem4 25840

Description: Lemma 4 for frgrancvvdeq 25849. 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 3622	. 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 2541	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frgrancvvdeq.f	. . . . . . . 8 |- ( ph -> V FriendGrph E )
6		frisusgra 25799	. . . . . . . 8 |- ( V FriendGrph E -> V USGrph E )
7	5, 6	syl 17	. . . . . . 7 |- ( ph -> V USGrph E )
8		nbgraisvtx 25238	. . . . . . 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 225	. . . . 5 |- ( ph -> ( x e. D -> x e. V ) )
11	10	imp 436	. . . 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 25837	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18	5	adantr 472	. . . . 5 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
19		frisusgranb 25804	. . . . 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 543	. . 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 3969	. . . . . . . . 9 |- ( a = x -> { a } = { x } )
23	22	difeq2d 3540	. . . . . . . 8 |- ( a = x -> ( V \ { a } ) = ( V \ { x } ) )
24		oveq2 6316	. . . . . . . . . . 11 |- ( a = x -> ( <. V , E >. Neighbors a ) = ( <. V , E >. Neighbors x ) )
25	24	ineq1d 3624	. . . . . . . . . 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 2473	. . . . . . . . 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 2892	. . . . . . . 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 2987	. . . . . . 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 3134	. . . . . 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 6316	. . . . . . . . . . . 12 |- ( b = Y -> ( <. V , E >. Neighbors b ) = ( <. V , E >. Neighbors Y ) )
31	30	ineq2d 3625	. . . . . . . . . . 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 2473	. . . . . . . . . 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 2892	. . . . . . . . 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 3134	. . . . . . . 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 3989	. . . . . . . . . . . . . . 15 |- n e. { n }
36		eleq2 2538	. . . . . . . . . . . . . . . . 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 3608	. . . . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . 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 25237	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { n , x } e. ran E ) )
43		prcom 4041	. . . . . . . . . . . . . . . . . . . . . . 23 |- { n , x } = { x , n }
44	43	eleq1i 2540	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { n , x } e. ran E <-> { x , n } e. ran E )
45	42, 44	syl6bb 269	. . . . . . . . . . . . . . . . . . . . 21 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { x , n } e. ran E ) )
46	45	biimpd 212	. . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 436	. . . . . . . . . . . . . . 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 2541	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( <. V , E >. Neighbors Y ) <-> n e. N )
54	53	biimpi 199	. . . . . . . . . . . . . . . . 17 |- ( n e. ( <. V , E >. Neighbors Y ) -> n e. N )
55	54	adantl 473	. . . . . . . . . . . . . . . 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 25839	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )
57		preq2 4043	. . . . . . . . . . . . . . . . . 18 |- ( y = n -> { x , y } = { x , n } )
58	57	eleq1d 2533	. . . . . . . . . . . . . . . . 17 |- ( y = n -> ( { x , y } e. ran E <-> { x , n } e. ran E ) )
59	58	riota2 6292	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 2477	. . . . . . . . . . . 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 3971	. . . . . . . . . . 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 2475	. . . . . . . . . . . 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 472	. . . . . . . . . . 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 240	. . . . . . . . . 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 441	. . . . . . . . 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 2869	. . . . . . . 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 442	. . . . . 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 447	. . . 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 436	. . 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 2518	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   e/ wnel 2642   A.wral 2756   E.wrex 2757   E!wreu 2758   \ cdif 3387   i^i cin 3389   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   ran crn 4840   iota_crio 6269  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224   FriendGrph cfrgra 25795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-frgra 25796

This theorem is referenced by:  frgrancvvdeqlem6  25842