Theorem refrelid 38026

Description: Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)

Assertion

Ref	Expression
refrelid	⊢ RefRel I

Proof of Theorem refrelid

Step	Hyp	Ref	Expression
1		ssid 4004	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5832	. 2 ⊢ Rel I
3		df-refrel 38016	. 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
4	1, 2, 3	mpbir2an 709	1 ⊢ RefRel I

Syntax hints:   ∩ cin 3948   ⊆ wss 3949   I cid 5579   × cxp 5680  dom cdm 5682  ran crn 5683  Rel wrel 5687   RefRel wrefrel 37687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-refrel 38016

This theorem is referenced by: (None)