Theorem funi 6466

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6560. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5736	. 2 ⊢ Rel I
2		relcnv 6012	. . . . 5 ⊢ Rel ◡ I
3		coi2 6167	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6045	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2766	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3979	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6435	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 708	1 ⊢ Fun I

Syntax hints:   = wceq 1539   ⊆ wss 3887   I cid 5488  ^◡ccnv 5588   ∘ ccom 5593  Rel wrel 5594  Fun wfun 6427

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-fun 6435

This theorem is referenced by:  cnvresid  6513  idfn  6560  fnresiOLD  6562  fvi  6844  resiexd  7092  ssdomg  8786  residfi  9100  bj-funidres  35322  tendo02  38801