Theorem funi 6610

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6708. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5850	. 2 ⊢ Rel I
2		relcnv 6134	. . . . 5 ⊢ Rel ◡ I
3		coi2 6294	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6173	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2768	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4069	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6575	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 710	1 ⊢ Fun I

Syntax hints:   = wceq 1537   ⊆ wss 3976   I cid 5592  ^◡ccnv 5699   ∘ ccom 5704  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  cnvresid  6657  idfn  6708  fvi  6998  resiexd  7253  ssdomg  9060  residfi  9406  bj-funidres  37117  tendo02  40744  grimidvtxedg  47760