Theorem refrelid 39113

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 3961	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5804	. 2 ⊢ Rel I
3		df-refrel 39103	. 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
4	1, 2, 3	mpbir2an 723	1 ⊢ RefRel I

Syntax hints:   ∩ cin 3906   ⊆ wss 3907   I cid 5546   × cxp 5650  dom cdm 5652  ran crn 5653  Rel wrel 5657   RefRel wrefrel 38700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-refrel 39103

This theorem is referenced by: (None)