Theorem funi 6530

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6626. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5782	. 2 ⊢ Rel I
2		relcnv 6069	. . . . 5 ⊢ Rel ◡ I
3		coi2 6228	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6105	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2759	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3982	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6500	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 712	1 ⊢ Fun I

Syntax hints:   = wceq 1542   ⊆ wss 3889   I cid 5525  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636  Fun wfun 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6500

This theorem is referenced by:  cnvresid  6577  idfn  6626  f1oi  6818  fvi  6916  resiexd  7171  ssdomg  8947  residfi  9248  bj-funidres  37465  tendo02  41233  grimidvtxedg  48361