Theorem funi 6600

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6697. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5839	. 2 ⊢ Rel I
2		relcnv 6125	. . . . 5 ⊢ Rel ◡ I
3		coi2 6285	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6164	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2763	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4056	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6565	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1537   ⊆ wss 3963   I cid 5582  ^◡ccnv 5688   ∘ ccom 5693  Rel wrel 5694  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565

This theorem is referenced by:  cnvresid  6647  idfn  6697  fvi  6985  resiexd  7236  ssdomg  9039  residfi  9376  bj-funidres  37134  tendo02  40770  grimidvtxedg  47814