Theorem funi 6564

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6659. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5804	. 2 ⊢ Rel I
2		relcnv 6098	. . . . 5 ⊢ Rel ◡ I
3		coi2 6258	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5863	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2784	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3991	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6533	. 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 3899   I cid 5545  ^◡ccnv 5650   ∘ ccom 5655  Rel wrel 5656  Fun wfun 6525

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6533

This theorem is used by:  cnvresid  6611  idfn  6659  f1oi  6855  fvi  6953  resiexd  7214  ssdomg  9011  residfi  9311  bj-funidres  38040  tendo02  41812  grimidvtxedg  48927