Theorem fndmu 5299

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 5297	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		fndm 5297	. 2 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
3	1, 2	sylan9req 2224	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  dom cdm 4611   Fn wfn 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-fn 5201

This theorem is referenced by:  fodmrnu  5428  tfrlemisucaccv  6304  tfr1onlemsucaccv  6320  tfrcllemsucaccv  6333  0fz1  10001