Theorem relsubgr 29630

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 29629	. 2 ⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
2	1	relopabiv 5806	1 ⊢ Rel SubGraph

This proof depends on syntax axioms:   ∧ w3a 1102   = wceq 1569   ⊆ wss 3904  𝒫 cpw 4561  dom cdm 5660   ↾ cres 5662  Rel wrel 5665  ‘cfv 6536  Vtxcvtx 29357  iEdgciedg 29358  Edgcedg 29408   SubGraph csubgr 29628

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-opab 5173  df-xp 5666  df-rel 5667  df-subgr 29629

This theorem is used by:  subgrv  29631