Theorem funi 5635

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 4984	. 2 |- Rel _I
2		relcnv 5229	. . . . 5 |- Rel `' _I
3		coi2 5375	. . . . 5 |- ( Rel `' _I -> ( _I o. `' _I ) = `' _I )
4	2, 3	ax-mp 5	. . . 4 |- ( _I o. `' _I ) = `' _I
5		cnvi 5262	. . . 4 |- `' _I = _I
6	4, 5	eqtri 2484	. . 3 |- ( _I o. `' _I ) = _I
7	6	eqimssi 3498	. 2 |- ( _I o. `' _I ) C_ _I
8		df-fun 5607	. 2 |- ( Fun _I <-> ( Rel _I /\ ( _I o. `' _I ) C_ _I ) )
9	1, 7, 8	mpbir2an 936	1 |- Fun _I

Syntax hints:   = wceq 1455   C_ wss 3416   _I cid 4766   `'ccnv 4855   o. ccom 4860   Rel wrel 4861   Fun wfun 5599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-fun 5607

This theorem is referenced by:  cnvresid  5679  fnresi  5719  fvi  5950  resiexd  6161  ssdomg  7646  tendo02  34400  residfi  39177  usgresvm1  40124  usgresvm1ALT  40128