Theorem funi 5920

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 5249	. 2 ⊢ Rel I
2		relcnv 5503	. . . . 5 ⊢ Rel ◡ I
3		coi2 5652	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5537	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2644	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3659	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 5890	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 955	1 ⊢ Fun I

Syntax hints:   = wceq 1483   ⊆ wss 3574   I cid 5023  ^◡ccnv 5113   ∘ ccom 5118  Rel wrel 5119  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  cnvresid  5968  fnresi  6008  fvi  6255  resiexd  6480  ssdomg  8001  residfi  8247  tendo02  36075