Theorem refrelid 36639

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 3943	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5736	. 2 ⊢ Rel I
3		df-refrel 36630	. 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
4	1, 2, 3	mpbir2an 708	1 ⊢ RefRel I

Syntax hints:   ∩ cin 3886   ⊆ wss 3887   I cid 5488   × cxp 5587  dom cdm 5589  ran crn 5590  Rel wrel 5594   RefRel wrefrel 36339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-refrel 36630

This theorem is referenced by: (None)