Theorem f1odm 5418

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 5415	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5269	. 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 1335  dom cdm 4586   Fn wfn 5165  –1-1-onto→wf1o 5169

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 5173  df-f 5174  df-f1 5175  df-f1o 5177

This theorem is referenced by:  f1imacnv  5431  f1opw2  6026  xpcomco  6771  mapen  6791  ssenen  6796  phplem4  6800  phplem4on  6812  dif1en  6824  fiintim  6873  caseinl  7035  caseinr  7036  ctssdccl  7055  fihasheqf1oi  10662  hashfacen  10707  fisumss  11289