Theorem f1of1 5287

Description: A one-to-one onto mapping is a one-to-one mapping. (Contributed by set.mm contributors, 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 4795	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
2	1	simplbi 446	1 ⊢ (F:A–1-1-onto→B → F:A–1-1→B)

This proof depends on syntax axioms:   → wi 4  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-f1o 4795

This theorem is used by:  f1of  5288  isoini2  5499  f1oiso  5500  swapres  5513  enpw1  6063  enpw1pw  6076  ncdisjun  6137  dflec3  6222  nclenc  6223  lenc  6224