Theorem funi 6551

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6649. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5792	. 2 ⊢ Rel I
2		relcnv 6078	. . . . 5 ⊢ Rel ◡ I
3		coi2 6239	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6117	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2753	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4010	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6516	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1540   ⊆ wss 3917   I cid 5535  ^◡ccnv 5640   ∘ ccom 5645  Rel wrel 5646  Fun wfun 6508

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-fun 6516

This theorem is referenced by:  cnvresid  6598  idfn  6649  fvi  6940  resiexd  7193  ssdomg  8974  residfi  9296  bj-funidres  37146  tendo02  40788  grimidvtxedg  47889