Theorem funi 5358

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

Assertion

Ref	Expression
funi	|- Fun _I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 4859	. 2 |- Rel _I
2		relcnv 5114	. . . . 5 |- Rel `' _I
3		coi2 5253	. . . . 5 |- ( Rel `' _I -> ( _I o. `' _I ) = `' _I )
4	2, 3	ax-mp 5	. . . 4 |- ( _I o. `' _I ) = `' _I
5		cnvi 5141	. . . 4 |- `' _I = _I
6	4, 5	eqtri 2252	. . 3 |- ( _I o. `' _I ) = _I
7	6	eqimssi 3283	. 2 |- ( _I o. `' _I ) C_ _I
8		df-fun 5328	. 2 |- ( Fun _I <-> ( Rel _I /\ ( _I o. `' _I ) C_ _I ) )
9	1, 7, 8	mpbir2an 950	1 |- Fun _I

Syntax hints:   = wceq 1397   C_ wss 3200   _I cid 4385   `'ccnv 4724   o. ccom 4729   Rel wrel 4730   Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  cnvresid  5404  fnresi  5450  fvi  5703  ssdomg  6951  residfi  7138  climshft2  11866