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Theorem funi 6585

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6683. (Contributed by NM, 30-Apr-1998.)

**Assertion**
Ref	Expression
funi	⊢ Fun I

**Proof of Theorem funi**
Step	Hyp	Ref	Expression
1		reli 5828	. 2 ⊢ Rel I
2		relcnv 6108	. . . . 5 ⊢ Rel ^◡ I
3		coi2 6267	. . . . 5 ⊢ (Rel ^◡ I → ( I ∘ ^◡ I ) = ^◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ^◡ I ) = ^◡ I
5		cnvi 6146	. . . 4 ⊢ ^◡ I = I
6	4, 5	eqtri 2756	. . 3 ⊢ ( I ∘ ^◡ I ) = I
7	6	eqimssi 4040	. 2 ⊢ ( I ∘ ^◡ I ) ⊆ I
8		df-fun 6550	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ^◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 710	1 ⊢ Fun I

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ⊆ wss 3947 I cid 5575 ^◡ccnv 5677 ∘ ccom 5682 Rel wrel 5683 Fun wfun 6542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-fun 6550

This theorem is referenced by: cnvresid 6632 idfn 6683 fvi 6974 resiexd 7228 ssdomg 9021 residfi 9358 bj-funidres 36630 tendo02 40260 grimidvtxedg 47174

	
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