Theorem isomgrrel 46788

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 46787	. 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 5819	1 ⊢ Rel IsomGr

Syntax hints:   ∧ wa 394   = wceq 1539  ∃wex 1779  ∀wral 3059  dom cdm 5675   “ cima 5678  Rel wrel 5680  –1-1-onto→wf1o 6541  ‘cfv 6542  Vtxcvtx 28523  iEdgciedg 28524   IsomGr cisomgr 46785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-isomgr 46787

This theorem is referenced by:  isisomgr  46790