Theorem funi 6387

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6475. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5698	. 2 ⊢ Rel I
2		relcnv 5967	. . . . 5 ⊢ Rel ◡ I
3		coi2 6116	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6000	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2844	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4025	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6357	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 709	1 ⊢ Fun I

Syntax hints:   = wceq 1537   ⊆ wss 3936   I cid 5459  ^◡ccnv 5554   ∘ ccom 5559  Rel wrel 5560  Fun wfun 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-fun 6357

This theorem is referenced by:  cnvresid  6433  idfn  6475  fnresiOLD  6477  fvi  6740  resiexd  6979  ssdomg  8555  residfi  8805  bj-funidres  34446  tendo02  37938