Theorem funi 6548

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6646. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5789	. 2 ⊢ Rel I
2		relcnv 6075	. . . . 5 ⊢ Rel ◡ I
3		coi2 6236	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6114	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2752	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 4007	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6513	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 711	1 ⊢ Fun I

Syntax hints:   = wceq 1540   ⊆ wss 3914   I cid 5532  ^◡ccnv 5637   ∘ ccom 5642  Rel wrel 5643  Fun wfun 6505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-fun 6513

This theorem is referenced by:  cnvresid  6595  idfn  6646  fvi  6937  resiexd  7190  ssdomg  8971  residfi  9289  bj-funidres  37139  tendo02  40781  grimidvtxedg  47885