Theorem funi 6520

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6616. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5772	. 2 ⊢ Rel I
2		relcnv 6059	. . . . 5 ⊢ Rel ◡ I
3		coi2 6218	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6095	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2756	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3991	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6490	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1541   ⊆ wss 3898   I cid 5515  ^◡ccnv 5620   ∘ ccom 5625  Rel wrel 5626  Fun wfun 6482

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-fun 6490

This theorem is referenced by:  cnvresid  6567  idfn  6616  f1oi  6808  fvi  6906  resiexd  7158  ssdomg  8931  residfi  9231  bj-funidres  37218  tendo02  40909  grimidvtxedg  48012