Theorem idfn 6684

Description: The identity relation is a function on the universal class. See also funi 6586. (Contributed by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idfn	⊢ I Fn V

Proof of Theorem idfn

Step	Hyp	Ref	Expression
1		funi 6586	. 2 ⊢ Fun I
2		dmi 5924	. 2 ⊢ dom I = V
3		df-fn 6552	. 2 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
4	1, 2, 3	mpbir2an 709	1 ⊢ I Fn V

Syntax hints:   = wceq 1533  Vcvv 3461   I cid 5575  dom cdm 5678  Fun wfun 6543   Fn wfn 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6551  df-fn 6552

This theorem is referenced by:  fnresi  6685