Theorem f1odm 5371

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 5368	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5222	. 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 1331  dom cdm 4539   Fn wfn 5118  –1-1-onto→wf1o 5122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130

This theorem is referenced by:  f1imacnv  5384  f1opw2  5976  xpcomco  6720  mapen  6740  ssenen  6745  phplem4  6749  phplem4on  6761  dif1en  6773  fiintim  6817  caseinl  6976  caseinr  6977  ctssdccl  6996  fihasheqf1oi  10534  hashfacen  10579  fisumss  11161