Theorem f1odm 5508

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 5505	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5357	. 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 1364  dom cdm 4663   Fn wfn 5253  –1-1-onto→wf1o 5257

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 5261  df-f 5262  df-f1 5263  df-f1o 5265

This theorem is referenced by:  f1imacnv  5521  f1opw2  6129  xpcomco  6885  mapen  6907  ssenen  6912  phplem4  6916  phplem4on  6928  dif1en  6940  fiintim  6992  caseinl  7157  caseinr  7158  ctssdccl  7177  fihasheqf1oi  10879  hashfacen  10928  fisumss  11557