Theorem funi 6513

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6609. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5766	. 2 ⊢ Rel I
2		relcnv 6053	. . . . 5 ⊢ Rel ◡ I
3		coi2 6211	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6088	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2754	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3995	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6483	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1541   ⊆ wss 3902   I cid 5510  ^◡ccnv 5615   ∘ ccom 5620  Rel wrel 5621  Fun wfun 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-fun 6483

This theorem is referenced by:  cnvresid  6560  idfn  6609  fvi  6898  resiexd  7150  ssdomg  8922  residfi  9222  bj-funidres  37191  tendo02  40832  grimidvtxedg  47922