Theorem funi 5557

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 5076	. 2 |- Rel _I
2		relcnv 5315	. . . . 5 |- Rel `' _I
3		coi2 5463	. . . . 5 |- ( Rel `' _I -> ( _I o. `' _I ) = `' _I )
4	2, 3	ax-mp 5	. . . 4 |- ( _I o. `' _I ) = `' _I
5		cnvi 5350	. . . 4 |- `' _I = _I
6	4, 5	eqtri 2483	. . 3 |- ( _I o. `' _I ) = _I
7	6	eqimssi 3519	. 2 |- ( _I o. `' _I ) C_ _I
8		df-fun 5529	. 2 |- ( Fun _I <-> ( Rel _I /\ ( _I o. `' _I ) C_ _I ) )
9	1, 7, 8	mpbir2an 911	1 |- Fun _I

Syntax hints:   = wceq 1370   C_ wss 3437   _I cid 4740   `'ccnv 4948   o. ccom 4953   Rel wrel 4954   Fun wfun 5521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-fun 5529

This theorem is referenced by:  cnvresid  5597  fnresi  5637  fvi  5858  ssdomg  7466  tendo02  34770