Theorem refrelid 38634

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 3953	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5770	. 2 ⊢ Rel I
3		df-refrel 38624	. 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 3897   ⊆ wss 3898   I cid 5513   × cxp 5617  dom cdm 5619  ran crn 5620  Rel wrel 5624   RefRel wrefrel 38248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-ss 3915  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-refrel 38624

This theorem is referenced by: (None)