Theorem funi 6569

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6664. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5811	. 2 ⊢ Rel I
2		relcnv 6104	. . . . 5 ⊢ Rel ◡ I
3		coi2 6264	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5869	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2785	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3994	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6539	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 724	1 ⊢ Fun I

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3902   I cid 5553  ^◡ccnv 5658   ∘ ccom 5663  Rel wrel 5664  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-fun 6539

This theorem is used by:  cnvresid  6616  idfn  6664  f1oi  6860  fvi  6958  resiexd  7219  ssdomg  9010  residfi  9309  bj-funidres  37911  tendo02  41668  grimidvtxedg  48809