Theorem f1ovi 6336

Description: The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)

Assertion

Ref	Expression
f1ovi	⊢ I :V–1-1-onto→V

Proof of Theorem f1ovi

Step	Hyp	Ref	Expression
1		f1oi 6335	. 2 ⊢ ( I ↾ V):V–1-1-onto→V
2		reli 5405	. . . 4 ⊢ Rel I
3		dfrel3 5750	. . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I )
4	2, 3	mpbi 220	. . 3 ⊢ ( I ↾ V) = I
5		f1oeq1 6288	. . 3 ⊢ (( I ↾ V) = I → (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V))
6	4, 5	ax-mp 5	. 2 ⊢ (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V)
7	1, 6	mpbi 220	1 ⊢ I :V–1-1-onto→V

Syntax hints:   ↔ wb 196   = wceq 1632  Vcvv 3340   I cid 5173   ↾ cres 5268  Rel wrel 5271  –1-1-onto→wf1o 6048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056

This theorem is referenced by:  ncanth  6772