Theorem isomgrrel 46490

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 46489	. 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 5821	1 ⊢ Rel IsomGr

Syntax hints:   ∧ wa 397   = wceq 1542  ∃wex 1782  ∀wral 3062  dom cdm 5677   “ cima 5680  Rel wrel 5682  –1-1-onto→wf1o 6543  ‘cfv 6544  Vtxcvtx 28256  iEdgciedg 28257   IsomGr cisomgr 46487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-isomgr 46489

This theorem is referenced by:  isisomgr  46492