Theorem f1odm 5467

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

Assertion

Ref	Expression
f1odm	⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)

Proof of Theorem f1odm

Step	Hyp	Ref	Expression
1		f1ofn 5464	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5317	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   = wceq 1353  dom cdm 4628   Fn wfn 5213  –1-1-onto→wf1o 5217

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 5221  df-f 5222  df-f1 5223  df-f1o 5225

This theorem is referenced by:  f1imacnv  5480  f1opw2  6079  xpcomco  6828  mapen  6848  ssenen  6853  phplem4  6857  phplem4on  6869  dif1en  6881  fiintim  6930  caseinl  7092  caseinr  7093  ctssdccl  7112  fihasheqf1oi  10769  hashfacen  10818  fisumss  11402