Theorem efmndmgm

Description: The monoid of endofunctions on a class A is a magma. (Contributed by AV, 28-Jan-2024)

	Ref	Expression
Hypothesis	efmndmgm.g	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
Assertion	efmndmgm	$$⊢G\in \mathrm{Mgm}$$

Proof

Step	Hyp	Ref	Expression
1		efmndmgm.g	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
2		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
3		eqid	$$⊢{+}_G={+}_G$$
4	1 2 3	efmndcl	$$⊢\left(f\in {\mathrm{Base}}_G\wedge g\in {\mathrm{Base}}_G\right)\to f{+}_{G}g\in {\mathrm{Base}}_G$$
5	4	rgen2	$$⊢\forall f\in {\mathrm{Base}}_G\forall g\in {\mathrm{Base}}_{G}f{+}_{G}g\in {\mathrm{Base}}_G$$
6	1	fvexi	$$⊢G\in \mathrm{V}$$
7	2 3	ismgm	$$⊢G\in \mathrm{V}\to \left(G\in \mathrm{Mgm}\leftrightarrow \forall f\in {\mathrm{Base}}_G\forall g\in {\mathrm{Base}}_{G}f{+}_{G}g\in {\mathrm{Base}}_G\right)$$
8	6 7	ax-mp	$$⊢G\in \mathrm{Mgm}\leftrightarrow \forall f\in {\mathrm{Base}}_G\forall g\in {\mathrm{Base}}_{G}f{+}_{G}g\in {\mathrm{Base}}_G$$
9	5 8	mpbir	$$⊢G\in \mathrm{Mgm}$$