Theorem f1dm 5606

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 5603	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5483	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  dom cdm 4776   Fn wfn 5375  –1-1→wf1 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-fn 5383  df-f 5384  df-f1 5385

This theorem is used by:  fun11iun  5663  tposf12  6544  f1dmvrnfibi  7263  f1vrnfibi  7264  exmidfodomrlemim  7560  hmeoimaf1o  15552  uspgr1edc  16698  exmidsbthrlem  17285