Theorem isomgrrel 43380

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 43379	. 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	relopabi 5540	1 ⊢ Rel IsomGr

Syntax hints:   ∧ wa 387   = wceq 1507  ∃wex 1742  ∀wral 3082  dom cdm 5403   “ cima 5406  Rel wrel 5408  –1-1-onto→wf1o 6184  ‘cfv 6185  Vtxcvtx 26496  iEdgciedg 26497   IsomGr cisomgr 43377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-opab 4988  df-xp 5409  df-rel 5410  df-isomgr 43379

This theorem is referenced by:  isisomgr  43382