Theorem wrdeq

Description: Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016)

	Ref	Expression
Assertion	wrdeq	$$⊢S=T\to \mathrm{Word}S=\mathrm{Word}T$$

Proof

Step	Hyp	Ref	Expression
1		sswrd	$$⊢S\subseteq T\to \mathrm{Word}S\subseteq \mathrm{Word}T$$
2		sswrd	$$⊢T\subseteq S\to \mathrm{Word}T\subseteq \mathrm{Word}S$$
3	1 2	anim12i	$$⊢\left(S\subseteq T\wedge T\subseteq S\right)\to \left(\mathrm{Word}S\subseteq \mathrm{Word}T\wedge \mathrm{Word}T\subseteq \mathrm{Word}S\right)$$
4		eqss	$$⊢S=T\leftrightarrow \left(S\subseteq T\wedge T\subseteq S\right)$$
5		eqss	$$⊢\mathrm{Word}S=\mathrm{Word}T\leftrightarrow \left(\mathrm{Word}S\subseteq \mathrm{Word}T\wedge \mathrm{Word}T\subseteq \mathrm{Word}S\right)$$
6	3 4 5	3imtr4i	$$⊢S=T\to \mathrm{Word}S=\mathrm{Word}T$$