Theorem mgmhmrcl

Description: Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020)

	Ref	Expression
Assertion	mgmhmrcl	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) )

Proof

Step	Hyp	Ref	Expression
1		df-mgmhm	⊢ MgmHom = ( 𝑠 ∈ Mgm , 𝑡 ∈ Mgm ↦ { 𝑓 ∈ ( ( Base ‘ 𝑡 ) ↑m ( Base ‘ 𝑠 ) ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑠 ) ∀ 𝑦 ∈ ( Base ‘ 𝑠 ) ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑠 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑡 ) ( 𝑓 ‘ 𝑦 ) ) } )
2	1	elmpocl	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) )