Theorem refrelid 38518

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 4021	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5843	. 2 ⊢ Rel I
3		df-refrel 38508	. 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 3965   ⊆ wss 3966   I cid 5586   × cxp 5691  dom cdm 5693  ran crn 5694  Rel wrel 5698   RefRel wrefrel 38182

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-ss 3983  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-refrel 38508

This theorem is referenced by: (None)