Theorem uvtxaval 39509

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 39453	. . 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 5888	. . . . 5 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4		isuvtxa.v	. . . . 5 |- V = (Vtx ` G )
5	3, 4	syl6eqr 2514	. . . 4 |- ( g = G -> (Vtx ` g ) = V )
6	5	difeq1d 3562	. . . . 5 |- ( g = G -> ( (Vtx ` g ) \ { v } ) = ( V \ { v } ) )
7		oveq1 6322	. . . . . 6 |- ( g = G -> ( g NeighbVtx v ) = ( G NeighbVtx v ) )
8	7	eleq2d 2525	. . . . 5 |- ( g = G -> ( n e. ( g NeighbVtx v ) <-> n e. ( G NeighbVtx v ) ) )
9	6, 8	raleqbidv 3013	. . . 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 3052	. . 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 472	. 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 3066	. 2 |- ( G e. W -> G e. _V )
13		fvex 5898	. . . . 5 |- (Vtx ` G ) e. _V
14	4, 13	eqeltri 2536	. . . 4 |- V e. _V
15	14	rabex 4568	. . 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 5977	1 |- ( G e. W -> (UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) } )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   A.wral 2749   {crab 2753   _Vcvv 3057   \ cdif 3413   {csn 3980   |-> cmpt 4475   ` cfv 5601  (class class class)co 6315  Vtxcvtx 39147   NeighbVtx cnbgr 39447  UnivVtxcuvtxa 39448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6318  df-uvtxa 39453

This theorem is referenced by:  uvtxael  39510  uvtxaisvtx  39511  uvtxa0  39516  isuvtxa  39517  uvtxa01vtx0  39519  uvtxusgr  39525