Theorem funi 6572

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6667. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5803	. 2 ⊢ Rel I
2		relcnv 6099	. . . . 5 ⊢ Rel ◡ I
3		coi2 6265	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5862	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2783	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3968	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6540	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 724	1 ⊢ Fun I

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3895   I cid 5544  ^◡ccnv 5649   ∘ ccom 5654  Rel wrel 5655  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-fun 6540

This theorem is used by:  cnvresid  6619  idfn  6667  f1oi  6863  fvi  6961  resiexd  7222  ssdomg  9030  residfi  9330  bj-funidres  38165  tendo02  41937  grimidvtxedg  49076