Theorem idfn 6676

Description: The identity relation is a function on the universal class. See also funi 6578. (Contributed by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idfn	⊢ I Fn V

Proof of Theorem idfn

Step	Hyp	Ref	Expression
1		funi 6578	. 2 ⊢ Fun I
2		dmi 5920	. 2 ⊢ dom I = V
3		df-fn 6544	. 2 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
4	1, 2, 3	mpbir2an 710	1 ⊢ I Fn V

Syntax hints:   = wceq 1542  Vcvv 3475   I cid 5573  dom cdm 5676  Fun wfun 6535   Fn wfn 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6543  df-fn 6544

This theorem is referenced by:  fnresi  6677