Theorem relsubgr 16410

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 16409	. 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 4898	1 |- Rel SubGraph

Syntax hints:   /\ w3a 1009   = wceq 1402   C_ wss 3220   ~Pcpw 3685   dom cdm 4769   |` cres 4771   Rel wrel 4774   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212   SubGraph csubgr 16408

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775  df-rel 4776  df-subgr 16409

This theorem is referenced by:  subgrv  16411