Theorem funi 6356

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6447. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5662	. 2 ⊢ Rel I
2		relcnv 5934	. . . . 5 ⊢ Rel ◡ I
3		coi2 6083	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5967	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2821	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3973	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6326	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 710	1 ⊢ Fun I

Syntax hints:   = wceq 1538   ⊆ wss 3881   I cid 5424  ^◡ccnv 5518   ∘ ccom 5523  Rel wrel 5524  Fun wfun 6318

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-fun 6326

This theorem is referenced by:  cnvresid  6403  idfn  6447  fnresiOLD  6449  fvi  6715  resiexd  6956  ssdomg  8538  residfi  8789  bj-funidres  34566  tendo02  38083