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Metamath Proof Explorer

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 ) )