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 5814	. 2 ⊢ Rel I
2		relcnv 6107	. . . . 5 ⊢ Rel ◡ I
3		coi2 6266	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5872	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2792	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4005	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6539	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 723	1 ⊢ Fun I

Syntax hints:   = wceq 1567   ⊆ wss 3913   I cid 5556  ^◡ccnv 5661   ∘ ccom 5666  Rel wrel 5667  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-fun 6539

This theorem is referenced by:  cnvresid  6616  idfn  6664  f1oi  6860  fvi  6958  resiexd  7215  ssdomg  8996  residfi  9294  bj-funidres  37682  tendo02  41450  grimidvtxedg  48538