Theorem f1odm 5436

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 5433	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5287	. 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 1343  dom cdm 4604   Fn wfn 5183  –1-1-onto→wf1o 5187

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 5191  df-f 5192  df-f1 5193  df-f1o 5195

This theorem is referenced by:  f1imacnv  5449  f1opw2  6044  xpcomco  6792  mapen  6812  ssenen  6817  phplem4  6821  phplem4on  6833  dif1en  6845  fiintim  6894  caseinl  7056  caseinr  7057  ctssdccl  7076  fihasheqf1oi  10701  hashfacen  10749  fisumss  11333