Theorem funi 5356

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 4857	. 2 ⊢ Rel I
2		relcnv 5112	. . . . 5 ⊢ Rel ◡ I
3		coi2 5251	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5139	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2250	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3281	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 5326	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 948	1 ⊢ Fun I

Syntax hints:   = wceq 1395   ⊆ wss 3198   I cid 4383  ^◡ccnv 4722   ∘ ccom 4727  Rel wrel 4728  Fun wfun 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-fun 5326

This theorem is referenced by:  cnvresid  5401  fnresi  5447  fvi  5699  ssdomg  6947  residfi  7133  climshft2  11860