Theorem funi 6524

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6620. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5776	. 2 ⊢ Rel I
2		relcnv 6063	. . . . 5 ⊢ Rel ◡ I
3		coi2 6222	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6099	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2763	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3982	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6494	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 717	1 ⊢ Fun I

Syntax hints:   = wceq 1547   ⊆ wss 3890   I cid 5519  ^◡ccnv 5624   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-fun 6494

This theorem is referenced by:  cnvresid  6571  idfn  6620  f1oi  6812  fvi  6910  resiexd  7167  ssdomg  8944  residfi  9245  bj-funidres  37518  tendo02  41286  grimidvtxedg  48383