Theorem rnghmsscmap2

Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020)

	Ref	Expression
Hypotheses	rnghmsscmap.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	rnghmsscmap.r	⊢ ( 𝜑 → 𝑅 = ( Rng ∩ 𝑈 ) )
Assertion	rnghmsscmap2	⊢ ( 𝜑 → ( RngHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		rnghmsscmap.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
2		rnghmsscmap.r	⊢ ( 𝜑 → 𝑅 = ( Rng ∩ 𝑈 ) )
3		ssidd	⊢ ( 𝜑 → 𝑅 ⊆ 𝑅 )
4		eqid	⊢ ( Base ‘ 𝑎 ) = ( Base ‘ 𝑎 )
5		eqid	⊢ ( Base ‘ 𝑏 ) = ( Base ‘ 𝑏 )
6	4 5	rnghmf	⊢ ( ℎ ∈ ( 𝑎 RngHom 𝑏 ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) )
7		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) )
8		fvex	⊢ ( Base ‘ 𝑏 ) ∈ V
9		fvex	⊢ ( Base ‘ 𝑎 ) ∈ V
10	8 9	pm3.2i	⊢ ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V )
11		elmapg	⊢ ( ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) )
12	10 11	mp1i	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) )
13	7 12	mpbird	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
14	13	ex	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) )
15	6 14	syl5	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ ∈ ( 𝑎 RngHom 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) )
16	15	ssrdv	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 RngHom 𝑏 ) ⊆ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
17		ovres	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( RngHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHom 𝑏 ) )
18	17	adantl	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHom 𝑏 ) )
19		eqidd	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
20		fveq2	⊢ ( 𝑦 = 𝑏 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑏 ) )
21		fveq2	⊢ ( 𝑥 = 𝑎 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑎 ) )
22	20 21	oveqan12rd	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
23	22	adantl	⊢ ( ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
24		simpl	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → 𝑎 ∈ 𝑅 )
25		simpr	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → 𝑏 ∈ 𝑅 )
26		ovexd	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ∈ V )
27	19 23 24 25 26	ovmpod	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
28	27	adantl	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
29	16 18 28	3sstr4d	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) )
30	29	ralrimivva	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) )
31		rnghmfn	⊢ RngHom Fn ( Rng × Rng )
32	31	a1i	⊢ ( 𝜑 → RngHom Fn ( Rng × Rng ) )
33		inss1	⊢ ( Rng ∩ 𝑈 ) ⊆ Rng
34	2 33	eqsstrdi	⊢ ( 𝜑 → 𝑅 ⊆ Rng )
35		xpss12	⊢ ( ( 𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng ) → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) )
36	34 34 35	syl2anc	⊢ ( 𝜑 → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) )
37		fnssres	⊢ ( ( RngHom Fn ( Rng × Rng ) ∧ ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) ) → ( RngHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) )
38	32 36 37	syl2anc	⊢ ( 𝜑 → ( RngHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) )
39		eqid	⊢ ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) )
40		ovex	⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V
41	39 40	fnmpoi	⊢ ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑅 × 𝑅 )
42	41	a1i	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑅 × 𝑅 ) )
43		incom	⊢ ( Rng ∩ 𝑈 ) = ( 𝑈 ∩ Rng )
44		inex1g	⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Rng ) ∈ V )
45	1 44	syl	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ∈ V )
46	43 45	eqeltrid	⊢ ( 𝜑 → ( Rng ∩ 𝑈 ) ∈ V )
47	2 46	eqeltrd	⊢ ( 𝜑 → 𝑅 ∈ V )
48	38 42 47	isssc	⊢ ( 𝜑 → ( ( RngHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ↔ ( 𝑅 ⊆ 𝑅 ∧ ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) ) )
49	3 30 48	mpbir2and	⊢ ( 𝜑 → ( RngHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )