Theorem funi 6598

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6696. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5836	. 2 ⊢ Rel I
2		relcnv 6122	. . . . 5 ⊢ Rel ◡ I
3		coi2 6283	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6161	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2765	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4044	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6563	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1540   ⊆ wss 3951   I cid 5577  ^◡ccnv 5684   ∘ ccom 5689  Rel wrel 5690  Fun wfun 6555

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563

This theorem is referenced by:  cnvresid  6645  idfn  6696  fvi  6985  resiexd  7236  ssdomg  9040  residfi  9378  bj-funidres  37152  tendo02  40789  grimidvtxedg  47876