Theorem idfn 6620

Description: The identity relation is a function on the universal class. See also funi 6524. (Contributed by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idfn	⊢ I Fn V

Proof of Theorem idfn

Step	Hyp	Ref	Expression
1		funi 6524	. 2 ⊢ Fun I
2		dmi 5870	. 2 ⊢ dom I = V
3		df-fn 6495	. 2 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
4	1, 2, 3	mpbir2an 711	1 ⊢ I Fn V

Syntax hints:   = wceq 1541  Vcvv 3440   I cid 5518  dom cdm 5624  Fun wfun 6486   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-fun 6494  df-fn 6495

This theorem is referenced by:  fnresi  6621