Theorem f1of1 5125

Description: A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Assertion

Ref	Expression
f1of1	⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)

Proof of Theorem f1of1

Step	Hyp	Ref	Expression
1		df-f1o 4909	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2	1	simplbi 259	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4  –1-1→wf1 4899  –onto→wfo 4900  –1-1-onto→wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99

This theorem depends on definitions:  df-bi 110  df-f1o 4909

This theorem is referenced by:  f1of  5126  f1oresrab  5329  f1ocnvfvrneq  5422  isores3  5455  isoini2  5458  f1oiso  5465  f1opw2  5706  tposf12  5884  enssdom  6242  phplem4  6318  phplem4on  6329  fidceq  6330