Theorem subgreldmiedg 16193

Description: An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)

Assertion

Ref	Expression
subgreldmiedg	|- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )

Proof of Theorem subgreldmiedg

Step	Hyp	Ref	Expression
1		eqid 2231	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2231	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2231	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16184	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		dmss 4936	. . . . 5 |- ( (iEdg ` S ) C_ (iEdg ` G ) -> dom (iEdg ` S ) C_ dom (iEdg ` G ) )
8	7	3ad2ant2 1046	. . . 4 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> dom (iEdg ` S ) C_ dom (iEdg ` G ) )
9	8	sseld 3227	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( X e. dom (iEdg ` S ) -> X e. dom (iEdg ` G ) ) )
10	6, 9	syl 14	. 2 |- ( S SubGraph G -> ( X e. dom (iEdg ` S ) -> X e. dom (iEdg ` G ) ) )
11	10	imp 124	1 |- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981   SubGraph csubgr 16177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-dm 4741  df-res 4743  df-iota 5293  df-fv 5341  df-subgr 16178

This theorem is referenced by:  subgruhgredgdm  16194  subumgredg2en  16195  subupgr  16197