Theorem idfn 6663

Description: The identity relation is a function on the universal class. See also funi 6568. (Contributed by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idfn	⊢ I Fn V

Proof of Theorem idfn

Step	Hyp	Ref	Expression
1		funi 6568	. 2 ⊢ Fun I
2		dmi 5911	. 2 ⊢ dom I = V
3		df-fn 6539	. 2 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
4	1, 2, 3	mpbir2an 723	1 ⊢ I Fn V

Syntax hints:   = wceq 1570  Vcvv 3455   I cid 5555  dom cdm 5661  Fun wfun 6530   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-fun 6538  df-fn 6539

This theorem is referenced by:  fnresi  6664