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	$$⊢\phi \to {\mathrm{Base}}_K={\mathrm{Base}}_L$$
	topnpropd.2	$$⊢\phi \to \mathrm{TopSet}\left(K\right)=\mathrm{TopSet}\left(L\right)$$
Assertion	topnpropd	$$⊢\phi \to \mathrm{TopOpen}\left(K\right)=\mathrm{TopOpen}\left(L\right)$$

Proof

Step	Hyp	Ref	Expression
1		topnpropd.1	$$⊢\phi \to {\mathrm{Base}}_K={\mathrm{Base}}_L$$
2		topnpropd.2	$$⊢\phi \to \mathrm{TopSet}\left(K\right)=\mathrm{TopSet}\left(L\right)$$
3	2 1	oveq12d	$$⊢\phi \to \mathrm{TopSet}\left(K\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_K=\mathrm{TopSet}\left(L\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_L$$
4		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
5		eqid	$$⊢\mathrm{TopSet}\left(K\right)=\mathrm{TopSet}\left(K\right)$$
6	4 5	topnval	$$⊢\mathrm{TopSet}\left(K\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_K=\mathrm{TopOpen}\left(K\right)$$
7		eqid	$$⊢{\mathrm{Base}}_L={\mathrm{Base}}_L$$
8		eqid	$$⊢\mathrm{TopSet}\left(L\right)=\mathrm{TopSet}\left(L\right)$$
9	7 8	topnval	$$⊢\mathrm{TopSet}\left(L\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_L=\mathrm{TopOpen}\left(L\right)$$
10	3 6 9	3eqtr3g	$$⊢\phi \to \mathrm{TopOpen}\left(K\right)=\mathrm{TopOpen}\left(L\right)$$