Theorem refrelid 34590

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 3761	. 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2		reli 5401	. 2 ⊢ Rel I
3		df-refrel 34581	. 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
4	1, 2, 3	mpbir2an 993	1 ⊢ RefRel I

Syntax hints:   ∩ cin 3710   ⊆ wss 3711   I cid 5169   × cxp 5260  dom cdm 5262  ran crn 5263  Rel wrel 5267   RefRel wrefrel 34298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4861  df-id 5170  df-xp 5268  df-rel 5269  df-refrel 34581

This theorem is referenced by: (None)