Theorem funi 6532

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6628. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5783	. 2 ⊢ Rel I
2		relcnv 6071	. . . . 5 ⊢ Rel ◡ I
3		coi2 6230	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 6107	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2760	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3996	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6502	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 712	1 ⊢ Fun I

Syntax hints:   = wceq 1542   ⊆ wss 3903   I cid 5526  ^◡ccnv 5631   ∘ ccom 5636  Rel wrel 5637  Fun wfun 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-fun 6502

This theorem is referenced by:  cnvresid  6579  idfn  6628  f1oi  6820  fvi  6918  resiexd  7172  ssdomg  8949  residfi  9250  bj-funidres  37403  tendo02  41160  grimidvtxedg  48242