Theorem fndmu 6105

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 6103	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		fndm 6103	. 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3	1, 2	sylan9req 2779	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596  dom cdm 5218   Fn wfn 5996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818  df-cleq 2717  df-fn 6004

This theorem is referenced by:  fodmrnu  6236  0fz1  12475  lmodfopnelem1  19022  grporn  27605  hon0  28882  2ffzoeq  41765