Theorem upgrres1lem2 26203

Description: Lemma 2 for upgrres1 26205. (Contributed by AV, 7-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	|- V = (Vtx ` G )
upgrres1.e	|- E = (Edg ` G )
upgrres1.f	|- F = { e e. E | N e/ e }
upgrres1.s	|- S = <. ( V \ { N } ) , ( _I |` F ) >.

Assertion

Ref	Expression
upgrres1lem2	|- (Vtx ` S ) = ( V \ { N } )

Distinct variable groups:   e, E   e, G   e, N   e, V

Allowed substitution hints:   S( e)   F( e)

Proof of Theorem upgrres1lem2

Step	Hyp	Ref	Expression
1		upgrres1.s	. . 3 |- S = <. ( V \ { N } ) , ( _I |` F ) >.
2	1	fveq2i 6194	. 2 |- (Vtx ` S ) = (Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. )
3		upgrres1.v	. . . 4 |- V = (Vtx ` G )
4		upgrres1.e	. . . 4 |- E = (Edg ` G )
5		upgrres1.f	. . . 4 |- F = { e e. E | N e/ e }
6	3, 4, 5	upgrres1lem1 26201	. . 3 |- ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V )
7		opvtxfv 25884	. . 3 |- ( ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V ) -> (Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } ) )
8	6, 7	ax-mp 5	. 2 |- (Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } )
9	2, 8	eqtri 2644	1 |- (Vtx ` S ) = ( V \ { N } )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   <.cop 4183   _I cid 5023   |` cres 5116   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939

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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  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-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-vtx 25876

This theorem is referenced by:  upgrres1  26205  umgrres1  26206  usgrres1  26207  nbupgrres  26266  nbupgruvtxres  26308  uvtxupgrres  26309  cusgrres  26344