Theorem idfn 6591

Description: The identity relation is a function on the universal class. See also funi 6495. (Contributed by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idfn	⊢ I Fn V

Proof of Theorem idfn

Step	Hyp	Ref	Expression
1		funi 6495	. 2 ⊢ Fun I
2		dmi 5843	. 2 ⊢ dom I = V
3		df-fn 6461	. 2 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
4	1, 2, 3	mpbir2an 709	1 ⊢ I Fn V

Syntax hints:   = wceq 1539  Vcvv 3437   I cid 5499  dom cdm 5600  Fun wfun 6452   Fn wfn 6453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-fun 6460  df-fn 6461

This theorem is referenced by:  fnresi  6592