Theorem vtxdginducedm1lem2 26436

Description: Lemma 2 for vtxdginducedm1 26439: the domain of 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
vtxdginducedm1lem2	|- dom (iEdg ` S ) = I

Distinct variable group:   i, E

Allowed substitution hints:   P( i)   S( i)   G( i)   I( i)   K( i)   N( i)   V( i)

Proof of Theorem vtxdginducedm1lem2

Step	Hyp	Ref	Expression
1		vtxdginducedm1.v	. . . . 5 |- V = (Vtx ` G )
2		vtxdginducedm1.e	. . . . 5 |- E = (iEdg ` G )
3		vtxdginducedm1.k	. . . . 5 |- K = ( V \ { N } )
4		vtxdginducedm1.i	. . . . 5 |- I = { i e. dom E | N e/ ( E ` i ) }
5		vtxdginducedm1.p	. . . . 5 |- P = ( E |` I )
6		vtxdginducedm1.s	. . . . 5 |- S = <. K , P >.
7	1, 2, 3, 4, 5, 6	vtxdginducedm1lem1 26435	. . . 4 |- (iEdg ` S ) = P
8	7, 5	eqtri 2644	. . 3 |- (iEdg ` S ) = ( E |` I )
9	8	dmeqi 5325	. 2 |- dom (iEdg ` S ) = dom ( E |` I )
10		ssrab2 3687	. . . 4 |- { i e. dom E | N e/ ( E ` i ) } C_ dom E
11	4, 10	eqsstri 3635	. . 3 |- I C_ dom E
12		ssdmres 5420	. . 3 |- ( I C_ dom E <-> dom ( E |` I ) = I )
13	11, 12	mpbi 220	. 2 |- dom ( E |` I ) = I
14	9, 13	eqtri 2644	1 |- dom (iEdg ` S ) = I

Syntax hints:   = wceq 1483   e/ wnel 2897   {crab 2916   \ cdif 3571   C_ wss 3574   {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:  vtxdginducedm1  26439