Theorem fndmu 6642

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 6638	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		fndm 6638	. 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3	1, 2	sylan9req 2819	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  dom cdm 5661   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-fn 6539

This theorem is referenced by:  fodmrnu  6800  0fz1  13567  lmodfopnelem1  21019  grporn  30873  hon0  32145  2ffzoeq  48065  homf0  49787  funchomf  49875