Theorem refrelid 37380

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 4003	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5824	. 2 ⊢ Rel I
3		df-refrel 37370	. 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 3946   ⊆ wss 3947   I cid 5572   × cxp 5673  dom cdm 5675  ran crn 5676  Rel wrel 5680   RefRel wrefrel 37037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-refrel 37370

This theorem is referenced by: (None)