Theorem rnghmsscmap

Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rnghmsscmap.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
2		rnghmsscmap.r	⊢ ( 𝜑 → 𝑅 = ( Rng ∩ 𝑈 ) )
3		inss2	⊢ ( Rng ∩ 𝑈 ) ⊆ 𝑈
4	2 3	eqsstrdi	⊢ ( 𝜑 → 𝑅 ⊆ 𝑈 )
5		eqid	⊢ ( Base ‘ 𝑎 ) = ( Base ‘ 𝑎 )
6		eqid	⊢ ( Base ‘ 𝑏 ) = ( Base ‘ 𝑏 )
7	5 6	rnghmf	⊢ ( ℎ ∈ ( 𝑎 RngHomo 𝑏 ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) )
8		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) )
9		fvex	⊢ ( Base ‘ 𝑏 ) ∈ V
10		fvex	⊢ ( Base ‘ 𝑎 ) ∈ V
11	9 10	pm3.2i	⊢ ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V )
12		elmapg	⊢ ( ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) )
13	11 12	mp1i	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) )
14	8 13	mpbird	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
15	14	ex	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) )
16	7 15	syl5	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ ∈ ( 𝑎 RngHomo 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) )
17	16	ssrdv	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 RngHomo 𝑏 ) ⊆ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
18		ovres	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHomo 𝑏 ) )
19	18	adantl	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHomo 𝑏 ) )
20		eqidd	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
21		fveq2	⊢ ( 𝑦 = 𝑏 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑏 ) )
22		fveq2	⊢ ( 𝑥 = 𝑎 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑎 ) )
23	21 22	oveqan12rd	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
24	23	adantl	⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
25	4	sseld	⊢ ( 𝜑 → ( 𝑎 ∈ 𝑅 → 𝑎 ∈ 𝑈 ) )
26	25	com12	⊢ ( 𝑎 ∈ 𝑅 → ( 𝜑 → 𝑎 ∈ 𝑈 ) )
27	26	adantr	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝜑 → 𝑎 ∈ 𝑈 ) )
28	27	impcom	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → 𝑎 ∈ 𝑈 )
29	4	sseld	⊢ ( 𝜑 → ( 𝑏 ∈ 𝑅 → 𝑏 ∈ 𝑈 ) )
30	29	com12	⊢ ( 𝑏 ∈ 𝑅 → ( 𝜑 → 𝑏 ∈ 𝑈 ) )
31	30	adantl	⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝜑 → 𝑏 ∈ 𝑈 ) )
32	31	impcom	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → 𝑏 ∈ 𝑈 )
33		ovexd	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ∈ V )
34	20 24 28 32 33	ovmpod	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) )
35	17 19 34	3sstr4d	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) )
36	35	ralrimivva	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) )
37		rnghmfn	⊢ RngHomo Fn ( Rng × Rng )
38	37	a1i	⊢ ( 𝜑 → RngHomo Fn ( Rng × Rng ) )
39		inss1	⊢ ( Rng ∩ 𝑈 ) ⊆ Rng
40	2 39	eqsstrdi	⊢ ( 𝜑 → 𝑅 ⊆ Rng )
41		xpss12	⊢ ( ( 𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng ) → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) )
42	40 40 41	syl2anc	⊢ ( 𝜑 → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) )
43		fnssres	⊢ ( ( RngHomo Fn ( Rng × Rng ) ∧ ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) ) → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) )
44	38 42 43	syl2anc	⊢ ( 𝜑 → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) )
45		eqid	⊢ ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) )
46		ovex	⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V
47	45 46	fnmpoi	⊢ ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 )
48	47	a1i	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 ) )
49		elex	⊢ ( 𝑈 ∈ 𝑉 → 𝑈 ∈ V )
50	1 49	syl	⊢ ( 𝜑 → 𝑈 ∈ V )
51	44 48 50	isssc	⊢ ( 𝜑 → ( ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ↔ ( 𝑅 ⊆ 𝑈 ∧ ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) ) )
52	4 36 51	mpbir2and	⊢ ( 𝜑 → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )