Theorem frgrncvvdeqlem3 27165

Description: Lemma 3 for frgrncvvdeq 27173. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	|- V = (Vtx ` G )
frgrncvvdeq.e	|- E = (Edg ` G )
frgrncvvdeq.nx	|- D = ( G NeighbVtx X )
frgrncvvdeq.ny	|- N = ( G NeighbVtx Y )
frgrncvvdeq.x	|- ( ph -> X e. V )
frgrncvvdeq.y	|- ( ph -> Y e. V )
frgrncvvdeq.ne	|- ( ph -> X =/= Y )
frgrncvvdeq.xy	|- ( ph -> Y e/ D )
frgrncvvdeq.f	|- ( ph -> G e. FriendGraph )
frgrncvvdeq.a	|- A = ( x e. D |-> ( iota_ y e. N { x , y } e. E ) )

Assertion

Ref	Expression
frgrncvvdeqlem3	|- ( ( ph /\ x e. D ) -> { ( iota_ y e. N { x , y } e. E ) } = ( ( G NeighbVtx x ) i^i N ) )

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

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

Proof of Theorem frgrncvvdeqlem3

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrncvvdeq.ny	. . 3 |- N = ( G NeighbVtx Y )
2	1	ineq2i 3811	. 2 |- ( ( G NeighbVtx x ) i^i N ) = ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) )
3		frgrncvvdeq.f	. . . . 5 |- ( ph -> G e. FriendGraph )
4	3	adantr 481	. . . 4 |- ( ( ph /\ x e. D ) -> G e. FriendGraph )
5		frgrncvvdeq.nx	. . . . . . . 8 |- D = ( G NeighbVtx X )
6	5	eleq2i 2693	. . . . . . 7 |- ( x e. D <-> x e. ( G NeighbVtx X ) )
7		frgrusgr 27124	. . . . . . . 8 |- ( G e. FriendGraph -> G e. USGraph )
8		frgrncvvdeq.v1	. . . . . . . . . 10 |- V = (Vtx ` G )
9	8	nbgrisvtx 26255	. . . . . . . . 9 |- ( ( G e. USGraph /\ x e. ( G NeighbVtx X ) ) -> x e. V )
10	9	ex 450	. . . . . . . 8 |- ( G e. USGraph -> ( x e. ( G NeighbVtx X ) -> x e. V ) )
11	3, 7, 10	3syl 18	. . . . . . 7 |- ( ph -> ( x e. ( G NeighbVtx X ) -> x e. V ) )
12	6, 11	syl5bi 232	. . . . . 6 |- ( ph -> ( x e. D -> x e. V ) )
13	12	imp 445	. . . . 5 |- ( ( ph /\ x e. D ) -> x e. V )
14		frgrncvvdeq.y	. . . . . 6 |- ( ph -> Y e. V )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ x e. D ) -> Y e. V )
16		frgrncvvdeq.xy	. . . . . . 7 |- ( ph -> Y e/ D )
17		elnelne2 2908	. . . . . . 7 |- ( ( x e. D /\ Y e/ D ) -> x =/= Y )
18	16, 17	sylan2 491	. . . . . 6 |- ( ( x e. D /\ ph ) -> x =/= Y )
19	18	ancoms 469	. . . . 5 |- ( ( ph /\ x e. D ) -> x =/= Y )
20	13, 15, 19	3jca 1242	. . . 4 |- ( ( ph /\ x e. D ) -> ( x e. V /\ Y e. V /\ x =/= Y ) )
21		frgrncvvdeq.e	. . . . 5 |- E = (Edg ` G )
22	8, 21	frcond3 27133	. . . 4 |- ( G e. FriendGraph -> ( ( x e. V /\ Y e. V /\ x =/= Y ) -> E. n e. V ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } ) )
23	4, 20, 22	sylc 65	. . 3 |- ( ( ph /\ x e. D ) -> E. n e. V ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } )
24		vex 3203	. . . . . . . . . 10 |- n e. _V
25		elinsn 4245	. . . . . . . . . 10 |- ( ( n e. _V /\ ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } ) -> ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) )
26	24, 25	mpan 706	. . . . . . . . 9 |- ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } -> ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) )
27	21	nbusgreledg 26249	. . . . . . . . . . . . . . . . 17 |- ( G e. USGraph -> ( n e. ( G NeighbVtx x ) <-> { n , x } e. E ) )
28		prcom 4267	. . . . . . . . . . . . . . . . . 18 |- { n , x } = { x , n }
29	28	eleq1i 2692	. . . . . . . . . . . . . . . . 17 |- ( { n , x } e. E <-> { x , n } e. E )
30	27, 29	syl6bb 276	. . . . . . . . . . . . . . . 16 |- ( G e. USGraph -> ( n e. ( G NeighbVtx x ) <-> { x , n } e. E ) )
31	30	biimpd 219	. . . . . . . . . . . . . . 15 |- ( G e. USGraph -> ( n e. ( G NeighbVtx x ) -> { x , n } e. E ) )
32	3, 7, 31	3syl 18	. . . . . . . . . . . . . 14 |- ( ph -> ( n e. ( G NeighbVtx x ) -> { x , n } e. E ) )
33	32	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. D ) -> ( n e. ( G NeighbVtx x ) -> { x , n } e. E ) )
34	33	com12 32	. . . . . . . . . . . 12 |- ( n e. ( G NeighbVtx x ) -> ( ( ph /\ x e. D ) -> { x , n } e. E ) )
35	34	adantr 481	. . . . . . . . . . 11 |- ( ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) -> ( ( ph /\ x e. D ) -> { x , n } e. E ) )
36	35	imp 445	. . . . . . . . . 10 |- ( ( ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) /\ ( ph /\ x e. D ) ) -> { x , n } e. E )
37	1	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( G NeighbVtx Y ) = N
38	37	eleq2i 2693	. . . . . . . . . . . . 13 |- ( n e. ( G NeighbVtx Y ) <-> n e. N )
39	38	biimpi 206	. . . . . . . . . . . 12 |- ( n e. ( G NeighbVtx Y ) -> n e. N )
40	39	adantl 482	. . . . . . . . . . 11 |- ( ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) -> n e. N )
41		frgrncvvdeq.x	. . . . . . . . . . . 12 |- ( ph -> X e. V )
42		frgrncvvdeq.ne	. . . . . . . . . . . 12 |- ( ph -> X =/= Y )
43		frgrncvvdeq.a	. . . . . . . . . . . 12 |- A = ( x e. D |-> ( iota_ y e. N { x , y } e. E ) )
44	8, 21, 5, 1, 41, 14, 42, 16, 3, 43	frgrncvvdeqlem2 27164	. . . . . . . . . . 11 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. E )
45		preq2 4269	. . . . . . . . . . . . 13 |- ( y = n -> { x , y } = { x , n } )
46	45	eleq1d 2686	. . . . . . . . . . . 12 |- ( y = n -> ( { x , y } e. E <-> { x , n } e. E ) )
47	46	riota2 6633	. . . . . . . . . . 11 |- ( ( n e. N /\ E! y e. N { x , y } e. E ) -> ( { x , n } e. E <-> ( iota_ y e. N { x , y } e. E ) = n ) )
48	40, 44, 47	syl2an 494	. . . . . . . . . 10 |- ( ( ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) /\ ( ph /\ x e. D ) ) -> ( { x , n } e. E <-> ( iota_ y e. N { x , y } e. E ) = n ) )
49	36, 48	mpbid 222	. . . . . . . . 9 |- ( ( ( n e. ( G NeighbVtx x ) /\ n e. ( G NeighbVtx Y ) ) /\ ( ph /\ x e. D ) ) -> ( iota_ y e. N { x , y } e. E ) = n )
50	26, 49	sylan 488	. . . . . . . 8 |- ( ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( iota_ y e. N { x , y } e. E ) = n )
51	50	eqcomd 2628	. . . . . . 7 |- ( ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> n = ( iota_ y e. N { x , y } e. E ) )
52	51	sneqd 4189	. . . . . 6 |- ( ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> { n } = { ( iota_ y e. N { x , y } e. E ) } )
53		eqeq1 2626	. . . . . . 7 |- ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } -> ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } <-> { n } = { ( iota_ y e. N { x , y } e. E ) } ) )
54	53	adantr 481	. . . . . 6 |- ( ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } <-> { n } = { ( iota_ y e. N { x , y } e. E ) } ) )
55	52, 54	mpbird 247	. . . . 5 |- ( ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } /\ ( ph /\ x e. D ) ) -> ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } )
56	55	ex 450	. . . 4 |- ( ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } -> ( ( ph /\ x e. D ) -> ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } ) )
57	56	rexlimivw 3029	. . 3 |- ( E. n e. V ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { n } -> ( ( ph /\ x e. D ) -> ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } ) )
58	23, 57	mpcom 38	. 2 |- ( ( ph /\ x e. D ) -> ( ( G NeighbVtx x ) i^i ( G NeighbVtx Y ) ) = { ( iota_ y e. N { x , y } e. E ) } )
59	2, 58	syl5req 2669	1 |- ( ( ph /\ x e. D ) -> { ( iota_ y e. N { x , y } e. E ) } = ( ( G NeighbVtx x ) i^i N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   E!wreu 2914   _Vcvv 3200   i^i cin 3573   {csn 4177   {cpr 4179   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-frgr 27121

This theorem is referenced by:  frgrncvvdeqlem5  27167