Theorem funi 6570

Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6665. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
funi	⊢ Fun I

Proof of Theorem funi

Step	Hyp	Ref	Expression
1		reli 5815	. 2 ⊢ Rel I
2		relcnv 6108	. . . . 5 ⊢ Rel ◡ I
3		coi2 6267	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5873	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2786	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3998	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 6540	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 723	1 ⊢ Fun I

Syntax hints:   = wceq 1570   ⊆ wss 3906   I cid 5557  ^◡ccnv 5662   ∘ ccom 5667  Rel wrel 5668  Fun wfun 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-fun 6540

This theorem is referenced by:  cnvresid  6617  idfn  6665  f1oi  6861  fvi  6959  resiexd  7216  ssdomg  8998  residfi  9296  bj-funidres  37776  tendo02  41542  grimidvtxedg  48633