Theorem funi 6568

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6666. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5805	. 2 ⊢ Rel I
2		relcnv 6091	. . . . 5 ⊢ Rel ◡ I
3		coi2 6252	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6130	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2758	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4019	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6533	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1540   ⊆ wss 3926   I cid 5547  ^◡ccnv 5653   ∘ ccom 5658  Rel wrel 5659  Fun wfun 6525

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-fun 6533

This theorem is referenced by:  cnvresid  6615  idfn  6666  fvi  6955  resiexd  7208  ssdomg  9014  residfi  9350  bj-funidres  37169  tendo02  40806  grimidvtxedg  47898