Theorem f1dm 5424

Description: The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
f1dm	⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)

Proof of Theorem f1dm

Step	Hyp	Ref	Expression
1		f1fn 5421	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5313	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   = wceq 1353  dom cdm 4625   Fn wfn 5209  –1-1→wf1 5211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-fn 5217  df-f 5218  df-f1 5219

This theorem is referenced by:  fun11iun  5480  tposf12  6266  f1dmvrnfibi  6939  f1vrnfibi  6940  exmidfodomrlemim  7196  hmeoimaf1o  13676  exmidsbthrlem  14621