Theorem f1odm 5484

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 5481	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5334	. 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 4644   Fn wfn 5230  –1-1-onto→wf1o 5234

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 5238  df-f 5239  df-f1 5240  df-f1o 5242

This theorem is referenced by:  f1imacnv  5497  f1opw2  6099  xpcomco  6851  mapen  6873  ssenen  6878  phplem4  6882  phplem4on  6894  dif1en  6906  fiintim  6956  caseinl  7119  caseinr  7120  ctssdccl  7139  fihasheqf1oi  10798  hashfacen  10847  fisumss  11431