Theorem uvtxael 26288

Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)

Hypothesis

Ref	Expression
uvtxael.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
uvtxael	|- ( G e. W -> ( N e. (UnivVtx ` G ) <-> ( N e. V /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) ) )

Distinct variable groups:   n, G   n, N   n, V

Allowed substitution hint:   W( n)

Proof of Theorem uvtxael

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxael.v	. . . 4 |- V = (Vtx ` G )
2	1	uvtxaval 26287	. . 3 |- ( G e. W -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
3	2	eleq2d 2687	. 2 |- ( G e. W -> ( N e. (UnivVtx ` G ) <-> N e. { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } ) )
4		sneq 4187	. . . . 5 |- ( v = N -> { v } = { N } )
5	4	difeq2d 3728	. . . 4 |- ( v = N -> ( V \ { v } ) = ( V \ { N } ) )
6		oveq2 6658	. . . . 5 |- ( v = N -> ( G NeighbVtx v ) = ( G NeighbVtx N ) )
7	6	eleq2d 2687	. . . 4 |- ( v = N -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx N ) ) )
8	5, 7	raleqbidv 3152	. . 3 |- ( v = N -> ( A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) )
9	8	elrab 3363	. 2 |- ( N e. { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } <-> ( N e. V /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) )
10	3, 9	syl6bb 276	1 |- ( G e. W -> ( N e. (UnivVtx ` G ) <-> ( N e. V /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874   NeighbVtx cnbgr 26224  UnivVtxcuvtxa 26225

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-uvtxa 26230

This theorem is referenced by:  vtxnbuvtx  26291  uvtx2vtx1edg  26299  uvtx2vtx1edgb  26300  uvtxnbgrb  26302  iscplgrnb  26312  cplgr1v  26326  cusgrexi  26339  structtocusgr  26342