Theorem vtxdginducedm1lem1 26435

Description: Lemma 1 for vtxdginducedm1 26439: the edge function in the induced subgraph S of a pseudograph G obtained by removing one vertex N. (Contributed by AV, 16-Dec-2021.)

Hypotheses

Ref	Expression
vtxdginducedm1.v	|- V = (Vtx ` G )
vtxdginducedm1.e	|- E = (iEdg ` G )
vtxdginducedm1.k	|- K = ( V \ { N } )
vtxdginducedm1.i	|- I = { i e. dom E | N e/ ( E ` i ) }
vtxdginducedm1.p	|- P = ( E |` I )
vtxdginducedm1.s	|- S = <. K , P >.

Assertion

Ref	Expression
vtxdginducedm1lem1	|- (iEdg ` S ) = P

Proof of Theorem vtxdginducedm1lem1

Step	Hyp	Ref	Expression
1		vtxdginducedm1.s	. . 3 |- S = <. K , P >.
2	1	fveq2i 6194	. 2 |- (iEdg ` S ) = (iEdg ` <. K , P >. )
3		vtxdginducedm1.k	. . . 4 |- K = ( V \ { N } )
4		vtxdginducedm1.v	. . . . . 6 |- V = (Vtx ` G )
5	4	fvexi 6202	. . . . 5 |- V e. _V
6	5	difexi 4809	. . . 4 |- ( V \ { N } ) e. _V
7	3, 6	eqeltri 2697	. . 3 |- K e. _V
8		vtxdginducedm1.p	. . . 4 |- P = ( E |` I )
9		vtxdginducedm1.e	. . . . . 6 |- E = (iEdg ` G )
10	9	fvexi 6202	. . . . 5 |- E e. _V
11	10	resex 5443	. . . 4 |- ( E |` I ) e. _V
12	8, 11	eqeltri 2697	. . 3 |- P e. _V
13	7, 12	opiedgfvi 25890	. 2 |- (iEdg ` <. K , P >. ) = P
14	2, 13	eqtri 2644	1 |- (iEdg ` S ) = P

Syntax hints:   = wceq 1483   e/ wnel 2897   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875

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-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-2nd 7169  df-iedg 25877

This theorem is referenced by:  vtxdginducedm1lem2  26436  vtxdginducedm1lem3  26437  vtxdginducedm1fi  26440  finsumvtxdg2ssteplem4  26444