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 5775	. 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 2759	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3994	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6494	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1541   ⊆ wss 3901   I cid 5518  ^◡ccnv 5623   ∘ ccom 5628  Rel wrel 5629  Fun wfun 6486

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-fun 6494

This theorem is referenced by:  cnvresid  6571  idfn  6620  f1oi  6812  fvi  6910  resiexd  7162  ssdomg  8937  residfi  9238  bj-funidres  37356  tendo02  41047  grimidvtxedg  48131