Theorem funi 5407

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 4907	. 2 ⊢ Rel I
2		relcnv 5163	. . . . 5 ⊢ Rel ◡ I
3		coi2 5302	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5190	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2259	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3304	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 5377	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 955	1 ⊢ Fun I

Syntax hints:   = wceq 1402   ⊆ wss 3220   I cid 4431  ^◡ccnv 4771   ∘ ccom 4776  Rel wrel 4777  Fun wfun 5369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-fun 5377

This theorem is referenced by:  cnvresid  5453  fnresi  5499  fvi  5757  ssdomg  7059  residfi  7248  climshft2  12055