Theorem refrelid 38482

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 3986	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5816	. 2 ⊢ Rel I
3		df-refrel 38472	. 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
4	1, 2, 3	mpbir2an 711	1 ⊢ RefRel I

Syntax hints:   ∩ cin 3930   ⊆ wss 3931   I cid 5557   × cxp 5663  dom cdm 5665  ran crn 5666  Rel wrel 5670   RefRel wrefrel 38147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-ss 3948  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-refrel 38472

This theorem is referenced by: (None)