Theorem fndmu 6596

Description: A function has a unique domain. (Contributed by NM, 11-Aug-1994.)

Assertion

Ref	Expression
fndmu	⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Proof of Theorem fndmu

Step	Hyp	Ref	Expression
1		fndm 6592	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		fndm 6592	. 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3	1, 2	sylan9req 2789	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  dom cdm 5621   Fn wfn 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2725  df-fn 6492

This theorem is referenced by:  fodmrnu  6751  0fz1  13451  lmodfopnelem1  20840  grporn  30522  hon0  31794  2ffzoeq  47489  homf0  49170  funchomf  49258