Theorem funi 6450

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6544. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5725	. 2 ⊢ Rel I
2		relcnv 6001	. . . . 5 ⊢ Rel ◡ I
3		coi2 6156	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6034	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2766	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3975	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6420	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 707	1 ⊢ Fun I

Syntax hints:   = wceq 1539   ⊆ wss 3883   I cid 5479  ^◡ccnv 5579   ∘ ccom 5584  Rel wrel 5585  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-fun 6420

This theorem is referenced by:  cnvresid  6497  idfn  6544  fnresiOLD  6546  fvi  6826  resiexd  7074  ssdomg  8741  residfi  9030  bj-funidres  35249  tendo02  38728