Theorem funi 6389

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6477. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5700	. 2 ⊢ Rel I
2		relcnv 5969	. . . . 5 ⊢ Rel ◡ I
3		coi2 6118	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6002	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2846	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4027	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6359	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 709	1 ⊢ Fun I

Syntax hints:   = wceq 1537   ⊆ wss 3938   I cid 5461  ^◡ccnv 5556   ∘ ccom 5561  Rel wrel 5562  Fun wfun 6351

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 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-fun 6359

This theorem is referenced by:  cnvresid  6435  idfn  6477  fnresiOLD  6479  fvi  6742  resiexd  6981  ssdomg  8557  residfi  8807  bj-funidres  34445  tendo02  37925