Theorem frgrancvvdeqlem4 30535

Description: Lemma 4 for frgrancvvdeq 30544. 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 3546	. 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 2505	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frgrancvvdeq.f	. . . . . . . 8 |- ( ph -> V FriendGrph E )
6		frisusgra 30493	. . . . . . . 8 |- ( V FriendGrph E -> V USGrph E )
7	5, 6	syl 16	. . . . . . 7 |- ( ph -> V USGrph E )
8		nbgraisvtx 23261	. . . . . . 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 30532	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18	5	adantr 462	. . . . 5 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
19		frisusgranb 30498	. . . . 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 531	. . 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 3884	. . . . . . . . 9 |- ( a = x -> { a } = { x } )
23	22	difeq2d 3471	. . . . . . . 8 |- ( a = x -> ( V \ { a } ) = ( V \ { x } ) )
24		oveq2 6098	. . . . . . . . . . 11 |- ( a = x -> ( <. V , E >. Neighbors a ) = ( <. V , E >. Neighbors x ) )
25	24	ineq1d 3548	. . . . . . . . . 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 2449	. . . . . . . . 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 2734	. . . . . . . 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 2929	. . . . . . 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 3068	. . . . . 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 6098	. . . . . . . . . . . 12 |- ( b = Y -> ( <. V , E >. Neighbors b ) = ( <. V , E >. Neighbors Y ) )
31	30	ineq2d 3549	. . . . . . . . . . 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 2449	. . . . . . . . . 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 2734	. . . . . . . . 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 3068	. . . . . . . 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 3903	. . . . . . . . . . . . . . 15 |- n e. { n }
36		eleq2 2502	. . . . . . . . . . . . . . . . 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 2444	. . . . . . . . . . . . . . . 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 3536	. . . . . . . . . . . . . . . . 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 23260	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V USGrph E -> ( n e. ( <. V , E >. Neighbors x ) <-> { n , x } e. ran E ) )
43		prcom 3950	. . . . . . . . . . . . . . . . . . . . . . 23 |- { n , x } = { x , n }
44	43	eleq1i 2504	. . . . . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . 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 2445	. . . . . . . . . . . . . . . . . . 19 |- ( <. V , E >. Neighbors Y ) = N
53	52	eleq2i 2505	. . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . 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 30534	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )
57		preq2 3952	. . . . . . . . . . . . . . . . . 18 |- ( y = n -> { x , y } = { x , n } )
58	57	eleq1d 2507	. . . . . . . . . . . . . . . . 17 |- ( y = n -> ( { x , y } e. ran E <-> { x , n } e. ran E ) )
59	58	riota2 6073	. . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 2446	. . . . . . . . . . . 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 3886	. . . . . . . . . . 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 2447	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 2835	. . . . . . . 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 438	. . . 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 2486	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 1364   e. wcel 1761   =/= wne 2604   e/ wnel 2605   A.wral 2713   E.wrex 2714   E!wreu 2715   \ cdif 3322   i^i cin 3324   {csn 3874   {cpr 3876   <.cop 3880   class class class wbr 4289   e. cmpt 4347   ran crn 4837   iota_crio 6048  (class class class)co 6090   USGrph cusg 23183   Neighbors cnbgra 23248   FriendGrph cfrgra 30489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-usgra 23185  df-nbgra 23251  df-frgra 30490

This theorem is referenced by:  frgrancvvdeqlem6  30537