Theorem uvtxaval 26287

Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)

Hypothesis

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

Assertion

Ref	Expression
uvtxaval	|- ( G e. W -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )

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

Allowed substitution hints:   W( v, n)

Proof of Theorem uvtxaval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uvtxa 26230	. . 3 |- UnivVtx = ( g e. _V |-> { v e. (Vtx ` g ) | A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) } )
2	1	a1i 11	. 2 |- ( G e. W -> UnivVtx = ( g e. _V |-> { v e. (Vtx ` g ) | A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) } ) )
3		fveq2 6191	. . . . 5 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4		isuvtxa.v	. . . . 5 |- V = (Vtx ` G )
5	3, 4	syl6eqr 2674	. . . 4 |- ( g = G -> (Vtx ` g ) = V )
6	5	difeq1d 3727	. . . . 5 |- ( g = G -> ( (Vtx ` g ) \ { v } ) = ( V \ { v } ) )
7		oveq1 6657	. . . . . 6 |- ( g = G -> ( g NeighbVtx v ) = ( G NeighbVtx v ) )
8	7	eleq2d 2687	. . . . 5 |- ( g = G -> ( n e. ( g NeighbVtx v ) <-> n e. ( G NeighbVtx v ) ) )
9	6, 8	raleqbidv 3152	. . . 4 |- ( g = G -> ( A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) <-> A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
10	5, 9	rabeqbidv 3195	. . 3 |- ( g = G -> { v e. (Vtx ` g ) | A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) } = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
11	10	adantl 482	. 2 |- ( ( G e. W /\ g = G ) -> { v e. (Vtx ` g ) | A. n e. ( (Vtx ` g ) \ { v } ) n e. ( g NeighbVtx v ) } = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )
12		elex 3212	. 2 |- ( G e. W -> G e. _V )
13		fvex 6201	. . . . 5 |- (Vtx ` G ) e. _V
14	4, 13	eqeltri 2697	. . . 4 |- V e. _V
15	14	rabex 4813	. . 3 |- { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } e. _V
16	15	a1i 11	. 2 |- ( G e. W -> { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } e. _V )
17	2, 11, 12, 16	fvmptd 6288	1 |- ( G e. W -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   |-> cmpt 4729   ` 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:  uvtxael  26288  uvtxaisvtx  26289  uvtxa0  26294  isuvtxa  26295  uvtxa01vtx0  26297  uvtxusgr  26303