Theorem relsubgr 29783

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 29782	. 2 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
2	1	relopabiv 5794	1 ⊢ Rel SubGraph

This proof depends on syntax axioms:   ∧ w3a 1103   = wceq 1570   ⊆ wss 3898  𝒫 cpw 4556  dom cdm 5647   ↾ cres 5649  Rel wrel 5652  ‘cfv 6527  Vtxcvtx 29507  iEdgciedg 29508  Edgcedg 29558   SubGraph csubgr 29781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3915  df-opab 5167  df-xp 5653  df-rel 5654  df-subgr 29782

This theorem is used by:  subgrv  29784