Theorem fndmu 6646

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 6642	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		fndm 6642	. 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3	1, 2	sylan9req 2821	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  dom cdm 5663   Fn wfn 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-fn 6543

This theorem is used by:  fodmrnu  6804  0fz1  13584  lmodfopnelem1  21049  grporn  30920  hon0  32192  2ffzoeq  48098  homf0  49820  funchomf  49908