Theorem topnpropd

Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Hypotheses	topnpropd.1	⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) )
	topnpropd.2	⊢ ( 𝜑 → ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐿 ) )
Assertion	topnpropd	⊢ ( 𝜑 → ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐿 ) )

Proof

Step	Hyp	Ref	Expression
1		topnpropd.1	⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) )
2		topnpropd.2	⊢ ( 𝜑 → ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐿 ) )
3	2 1	oveq12d	⊢ ( 𝜑 → ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) = ( ( TopSet ‘ 𝐿 ) ↾t ( Base ‘ 𝐿 ) ) )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5		eqid	⊢ ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐾 )
6	4 5	topnval	⊢ ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) = ( TopOpen ‘ 𝐾 )
7		eqid	⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
8		eqid	⊢ ( TopSet ‘ 𝐿 ) = ( TopSet ‘ 𝐿 )
9	7 8	topnval	⊢ ( ( TopSet ‘ 𝐿 ) ↾t ( Base ‘ 𝐿 ) ) = ( TopOpen ‘ 𝐿 )
10	3 6 9	3eqtr3g	⊢ ( 𝜑 → ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐿 ) )