Theorem f1odm 5509

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 5506	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5358	. 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 4664   Fn wfn 5254  –1-1-onto→wf1o 5258

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 5262  df-f 5263  df-f1 5264  df-f1o 5266

This theorem is referenced by:  f1imacnv  5522  f1opw2  6130  xpcomco  6886  mapen  6908  ssenen  6913  phplem4  6917  phplem4on  6929  dif1en  6941  fiintim  6993  caseinl  7158  caseinr  7159  ctssdccl  7178  fihasheqf1oi  10881  hashfacen  10930  fisumss  11559