Theorem indisuni

Description: The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Assertion	indisuni	⊢ ( I ‘ 𝐴 ) = ∪ { ∅ , 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		indislem	⊢ { ∅ , ( I ‘ 𝐴 ) } = { ∅ , 𝐴 }
2		fvex	⊢ ( I ‘ 𝐴 ) ∈ V
3		indistopon	⊢ ( ( I ‘ 𝐴 ) ∈ V → { ∅ , ( I ‘ 𝐴 ) } ∈ ( TopOn ‘ ( I ‘ 𝐴 ) ) )
4	2 3	ax-mp	⊢ { ∅ , ( I ‘ 𝐴 ) } ∈ ( TopOn ‘ ( I ‘ 𝐴 ) )
5	1 4	eqeltrri	⊢ { ∅ , 𝐴 } ∈ ( TopOn ‘ ( I ‘ 𝐴 ) )
6	5	toponunii	⊢ ( I ‘ 𝐴 ) = ∪ { ∅ , 𝐴 }