Definition df-bj-topmgmhom

Description: Define the set of topological magma morphisms (continuous magma morphisms) between two topological magmas. If domain and codomain are topological semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. This definition is currently stated with topological monoid domain and codomain, since topological magmas are currently not defined in set.mm. (Contributed by BJ, 10-Feb-2022)

	Ref	Expression
Assertion	df-bj-topmgmhom	$$⊢\underset{TopMgm}{⟶}=\left(x\in \mathrm{TopMnd},y\in \mathrm{TopMnd}⟼\left(x\underset{Top}{⟶}y\right)\cap \left(x\underset{Mgm}{⟶}y\right)\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ctopmgmhom	$$class\underset{TopMgm}{⟶}$$
1		vx	$$setvarx$$
2		ctmd	$$class\mathrm{TopMnd}$$
3		vy	$$setvary$$
4	1	cv	$$setvarx$$
5		ctophom	$$class\underset{Top}{⟶}$$
6	3	cv	$$setvary$$
7	4 6 5	co	$$class\left(x\underset{Top}{⟶}y\right)$$
8		cmgmhom	$$class\underset{Mgm}{⟶}$$
9	4 6 8	co	$$class\left(x\underset{Mgm}{⟶}y\right)$$
10	7 9	cin	$$class\left(\left(x\underset{Top}{⟶}y\right)\cap \left(x\underset{Mgm}{⟶}y\right)\right)$$
11	1 3 2 2 10	cmpo	$$class\left(x\in \mathrm{TopMnd},y\in \mathrm{TopMnd}⟼\left(x\underset{Top}{⟶}y\right)\cap \left(x\underset{Mgm}{⟶}y\right)\right)$$
12	0 11	wceq	$$wff\underset{TopMgm}{⟶}=\left(x\in \mathrm{TopMnd},y\in \mathrm{TopMnd}⟼\left(x\underset{Top}{⟶}y\right)\cap \left(x\underset{Mgm}{⟶}y\right)\right)$$