Theorem funi 5309

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 4812	. 2 ⊢ Rel I
2		relcnv 5066	. . . . 5 ⊢ Rel ◡ I
3		coi2 5205	. . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I )
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I
5		cnvi 5093	. . . 4 ⊢ ◡ I = I
6	4, 5	eqtri 2227	. . 3 ⊢ ( I ∘ ◡ I ) = I
7	6	eqimssi 3251	. 2 ⊢ ( I ∘ ◡ I ) ⊆ I
8		df-fun 5279	. 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I ))
9	1, 7, 8	mpbir2an 945	1 ⊢ Fun I

Syntax hints:   = wceq 1373   ⊆ wss 3168   I cid 4340  ^◡ccnv 4679   ∘ ccom 4684  Rel wrel 4685  Fun wfun 5271

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-br 4049  df-opab 4111  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-fun 5279

This theorem is referenced by:  cnvresid  5354  fnresi  5400  fvi  5646  ssdomg  6880  residfi  7054  climshft2  11667