Theorem funi 6581

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6679. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5827	. 2 ⊢ Rel I
2		relcnv 6104	. . . . 5 ⊢ Rel ◡ I
3		coi2 6263	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6142	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2761	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4043	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6546	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 710	1 ⊢ Fun I

Syntax hints:   = wceq 1542   ⊆ wss 3949   I cid 5574  ^◡ccnv 5676   ∘ ccom 5681  Rel wrel 5682  Fun wfun 6538

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 5300  ax-nul 5307  ax-pr 5428

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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546

This theorem is referenced by:  cnvresid  6628  idfn  6679  fvi  6968  resiexd  7218  ssdomg  8996  residfi  9333  bj-funidres  36032  tendo02  39658