Theorem funi 6595

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6692. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5811	. 2 ⊢ Rel I
2		relcnv 6113	. . . . 5 ⊢ Rel ◡ I
3		coi2 6281	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5871	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2786	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3973	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6563	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 724	1 ⊢ Fun I

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900   I cid 5550  ^◡ccnv 5655   ∘ ccom 5660  Rel wrel 5661  Fun wfun 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-fun 6563

This theorem is used by:  cnvresidOLD  6644  idfn  6692  f1oi  6889  fvi  6987  resiexd  7248  ssdomg  9058  residfi  9358  bj-funidres  38277  tendo02  42047  grimidvtxedg  49198