Theorem f1ovi 6888

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⟶1-1ontoV

Proof of Theorem f1ovi

Step	Hyp	Ref	Expression
1		f1oi 6886	. 2 ⊢ ( I ↾ V):V⟶1-1ontoV
2		reli 5810	. . . 4 ⊢ Rel I
3		dfrel3 6205	. . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I )
4	2, 3	mpbi 233	. . 3 ⊢ ( I ↾ V) = I
5		f1oeq1 6834	. . 3 ⊢ (( I ↾ V) = I → (( I ↾ V):V⟶1-1ontoV ↔ I :V⟶1-1ontoV))
6	4, 5	ax-mp 5	. 2 ⊢ (( I ↾ V):V⟶1-1ontoV ↔ I :V⟶1-1ontoV)
7	1, 6	mpbi 233	1 ⊢ I :V⟶1-1ontoV

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3454   I cid 5549   ↾ cres 5657  Rel wrel 5660  ⟶1-1ontowf1o 6557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565

This theorem is used by:  ncanth  7398  nregmodelf1o  46200