Theorem f1f 5421

Description: A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Assertion

Ref	Expression
f1f	⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)

Proof of Theorem f1f

Step	Hyp	Ref	Expression
1		df-f1 5221	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
2	1	simplbi 274	1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4  ^◡ccnv 4625  Fun wfun 5210  ⟶wf 5212  –1-1→wf1 5213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-f1 5221

This theorem is referenced by:  f1rn  5422  f1fn  5423  f1ss  5427  f1ssres  5430  f1of  5461  dff1o5  5470  fsnd  5504  cocan1  5787  f1o2ndf1  6228  brdomg  6747  f1dom2g  6755  f1domg  6757  dom3d  6773  f1imaen2g  6792  2dom  6804  xpdom2  6830  dom0  6837  phplem4dom  6861  isinfinf  6896  infm  6903  updjudhcoinlf  7078  updjudhcoinrg  7079  casef1  7088  djudom  7091  difinfsnlem  7097  difinfsn  7098  fihashf1rn  10767  ennnfonelemrn  12419  reeff1o  14164  pwle2  14718