Theorem f1odm 5504

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 5501	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fndm 5353	. 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 4659   Fn wfn 5249  –1-1-onto→wf1o 5253

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 5257  df-f 5258  df-f1 5259  df-f1o 5261

This theorem is referenced by:  f1imacnv  5517  f1opw2  6124  xpcomco  6880  mapen  6902  ssenen  6907  phplem4  6911  phplem4on  6923  dif1en  6935  fiintim  6985  caseinl  7150  caseinr  7151  ctssdccl  7170  fihasheqf1oi  10858  hashfacen  10907  fisumss  11535