Theorem frgrncvvdeqlem1 27163

Description: Lemma 1 for frgrncvvdeq 27173. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 28-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
frgrncvvdeqlem1	|- ( ph -> X e/ N )

Proof of Theorem frgrncvvdeqlem1

Step	Hyp	Ref	Expression
1		frgrncvvdeq.xy	. . . 4 |- ( ph -> Y e/ D )
2		df-nel 2898	. . . . 5 |- ( Y e/ D <-> -. Y e. D )
3		frgrncvvdeq.nx	. . . . . 6 |- D = ( G NeighbVtx X )
4	3	eleq2i 2693	. . . . 5 |- ( Y e. D <-> Y e. ( G NeighbVtx X ) )
5	2, 4	xchbinx 324	. . . 4 |- ( Y e/ D <-> -. Y e. ( G NeighbVtx X ) )
6	1, 5	sylib 208	. . 3 |- ( ph -> -. Y e. ( G NeighbVtx X ) )
7		frgrncvvdeq.f	. . . 4 |- ( ph -> G e. FriendGraph )
8		nbgrsym 26265	. . . 4 |- ( G e. FriendGraph -> ( X e. ( G NeighbVtx Y ) <-> Y e. ( G NeighbVtx X ) ) )
9	7, 8	syl 17	. . 3 |- ( ph -> ( X e. ( G NeighbVtx Y ) <-> Y e. ( G NeighbVtx X ) ) )
10	6, 9	mtbird 315	. 2 |- ( ph -> -. X e. ( G NeighbVtx Y ) )
11		frgrncvvdeq.ny	. . . 4 |- N = ( G NeighbVtx Y )
12		neleq2 2903	. . . 4 |- ( N = ( G NeighbVtx Y ) -> ( X e/ N <-> X e/ ( G NeighbVtx Y ) ) )
13	11, 12	ax-mp 5	. . 3 |- ( X e/ N <-> X e/ ( G NeighbVtx Y ) )
14		df-nel 2898	. . 3 |- ( X e/ ( G NeighbVtx Y ) <-> -. X e. ( G NeighbVtx Y ) )
15	13, 14	bitri 264	. 2 |- ( X e/ N <-> -. X e. ( G NeighbVtx Y ) )
16	10, 15	sylibr 224	1 |- ( ph -> X e/ N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   {cpr 4179   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228

This theorem is referenced by:  frgrncvvdeqlem7  27169  frgrncvvdeqlem8  27170  frgrncvvdeqlem9  27171