Theorem isomgrrel 44766

Description: The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.)

Assertion

Ref	Expression
isomgrrel	⊢ Rel IsomGr

Proof of Theorem isomgrrel

Dummy variables 𝑓 𝑔 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-isomgr 44765	. 2 ⊢ IsomGr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))))}
2	1	relopabiv 5667	1 ⊢ Rel IsomGr

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781  ∀wral 3070  dom cdm 5528   “ cima 5531  Rel wrel 5533  –1-1-onto→wf1o 6339  ‘cfv 6340  Vtxcvtx 26902  iEdgciedg 26903   IsomGr cisomgr 44763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-opab 5099  df-xp 5534  df-rel 5535  df-isomgr 44765

This theorem is referenced by:  isisomgr  44768