Theorem funi 6557

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6653. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5803	. 2 ⊢ Rel I
2		relcnv 6096	. . . . 5 ⊢ Rel ◡ I
3		coi2 6254	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5861	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2788	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3999	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6527	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 723	1 ⊢ Fun I

Syntax hints:   = wceq 1563   ⊆ wss 3907   I cid 5545  ^◡ccnv 5650   ∘ ccom 5655  Rel wrel 5656  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6527

This theorem is referenced by:  cnvresid  6604  idfn  6653  f1oi  6849  fvi  6947  resiexd  7204  ssdomg  8985  residfi  9283  bj-funidres  37650  tendo02  41418  grimidvtxedg  48506