Theorem funi 6159

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

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5486	. 2 ⊢ Rel I
2		relcnv 5748	. . . . 5 ⊢ Rel ◡ I
3		coi2 5897	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5782	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2849	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3884	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6129	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 702	1 ⊢ Fun I

Syntax hints:   = wceq 1656   ⊆ wss 3798   I cid 5251  ^◡ccnv 5345   ∘ ccom 5350  Rel wrel 5351  Fun wfun 6121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-fun 6129

This theorem is referenced by:  cnvresid  6205  fnresi  6245  fvi  6506  resiexd  6741  ssdomg  8274  residfi  8522  tendo02  36861