Theorem f1dm 5465

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 5462	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5354	. 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 1364  dom cdm 4660   Fn wfn 5250  –1-1→wf1 5252

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 5258  df-f 5259  df-f1 5260

This theorem is referenced by:  fun11iun  5522  tposf12  6324  f1dmvrnfibi  7005  f1vrnfibi  7006  exmidfodomrlemim  7263  hmeoimaf1o  14493  exmidsbthrlem  15582