Theorem f1dm 5096

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 5093	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fndm 4998	. 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 1243  dom cdm 4345   Fn wfn 4897  –1-1→wf1 4899

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100

This theorem depends on definitions:  df-bi 110  df-fn 4905  df-f 4906  df-f1 4907

This theorem is referenced by:  fun11iun  5147  tposf12  5884