Theorem relsubgr 16105

Description: The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)

Assertion

Ref	Expression
relsubgr	|- Rel SubGraph

Proof of Theorem relsubgr

Dummy variables g s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subgr 16104	. 2 |- SubGraph = { <. s , g >. | ( (Vtx ` s ) C_ (Vtx ` g ) /\ (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) /\ (Edg ` s ) C_ ~P (Vtx ` s ) ) }
2	1	relopabiv 4853	1 |- Rel SubGraph

Syntax hints:   /\ w3a 1004   = wceq 1397   C_ wss 3200   ~Pcpw 3652   dom cdm 4725   |` cres 4727   Rel wrel 4730   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907   SubGraph csubgr 16103

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-opab 4151  df-xp 4731  df-rel 4732  df-subgr 16104

This theorem is referenced by:  subgrv  16106