Theorem f1odm 5446

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 5443	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5297	. 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 1348  dom cdm 4611   Fn wfn 5193  –1-1-onto→wf1o 5197

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 5201  df-f 5202  df-f1 5203  df-f1o 5205

This theorem is referenced by:  f1imacnv  5459  f1opw2  6055  xpcomco  6804  mapen  6824  ssenen  6829  phplem4  6833  phplem4on  6845  dif1en  6857  fiintim  6906  caseinl  7068  caseinr  7069  ctssdccl  7088  fihasheqf1oi  10722  hashfacen  10771  fisumss  11355