Theorem s2co

Description: Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2co.1	$$⊢\phi \to F:X⟶Y$$
	s2co.2	$$⊢\phi \to A\in X$$
	s2co.3	$$⊢\phi \to B\in X$$
Assertion	s2co	$$⊢\phi \to F\circ ⟨“AB”⟩=⟨“F\left(A\right)F\left(B\right)”⟩$$

Proof

Step	Hyp	Ref	Expression
1		s2co.1	$$⊢\phi \to F:X⟶Y$$
2		s2co.2	$$⊢\phi \to A\in X$$
3		s2co.3	$$⊢\phi \to B\in X$$
4		df-s2	$$⊢⟨“AB”⟩=⟨“A”⟩\mathrm{++}⟨“B”⟩$$
5	2	s1cld	$$⊢\phi \to ⟨“A”⟩\in \mathrm{Word}X$$
6		s1co	$$⊢\left(A\in X\wedge F:X⟶Y\right)\to F\circ ⟨“A”⟩=⟨“F\left(A\right)”⟩$$
7	2 1 6	syl2anc	$$⊢\phi \to F\circ ⟨“A”⟩=⟨“F\left(A\right)”⟩$$
8		df-s2	$$⊢⟨“F\left(A\right)F\left(B\right)”⟩=⟨“F\left(A\right)”⟩\mathrm{++}⟨“F\left(B\right)”⟩$$
9	4 5 3 1 7 8	cats1co	$$⊢\phi \to F\circ ⟨“AB”⟩=⟨“F\left(A\right)F\left(B\right)”⟩$$