Theorem rnghmsscmap 20538

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.)

Hypotheses

Ref	Expression
rnghmsscmap.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rnghmsscmap.r	⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈))

Assertion

Ref	Expression
rnghmsscmap	⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem rnghmsscmap

Dummy variables 𝑎 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnghmsscmap.r	. . 3 ⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈))
2		inss2 4186	. . 3 ⊢ (Rng ∩ 𝑈) ⊆ 𝑈
3	1, 2	eqsstrdi 3977	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
4		eqid 2730	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
5		eqid 2730	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
6	4, 5	rnghmf 20359	. . . . . 6 ⊢ (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
7		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
8		fvex 6830	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
9		fvex 6830	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
10	8, 9	pm3.2i 470	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11		elmapg 8758	. . . . . . . . 9 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
12	10, 11	mp1i 13	. . . . . . . 8 ⊢ (((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 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
14	13	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
15	6, 14	syl5 34	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
16	15	ssrdv 3938	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17		ovres 7507	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
18	17	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
19		eqidd 2731	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
20		fveq2 6817	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21		fveq2 6817	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
22	20, 21	oveqan12rd 7361	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
23	22	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
24	3	sseld 3931	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝑅 → 𝑎 ∈ 𝑈))
25	24	com12 32	. . . . . . 7 ⊢ (𝑎 ∈ 𝑅 → (𝜑 → 𝑎 ∈ 𝑈))
26	25	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑎 ∈ 𝑈))
27	26	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑎 ∈ 𝑈)
28	3	sseld 3931	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ 𝑅 → 𝑏 ∈ 𝑈))
29	28	com12 32	. . . . . . 7 ⊢ (𝑏 ∈ 𝑅 → (𝜑 → 𝑏 ∈ 𝑈))
30	29	adantl 481	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑏 ∈ 𝑈))
31	30	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑏 ∈ 𝑈)
32		ovexd 7376	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
33	19, 23, 27, 31, 32	ovmpod 7493	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
34	16, 18, 33	3sstr4d 3988	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
35	34	ralrimivva 3173	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36		rnghmfn 20350	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
38		inss1 4185	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
39	1, 38	eqsstrdi 3977	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Rng)
40		xpss12 5629	. . . . 5 ⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
41	39, 39, 40	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42		fnssres 6600	. . . 4 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
43	37, 41, 42	syl2anc 584	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45		ovex 7374	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
46	44, 45	fnmpoi 7997	. . . 4 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑈 × 𝑈)
47	46	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑈 × 𝑈))
48		rnghmsscmap.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
49		elex 3455	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
51	43, 47, 50	isssc 17719	. 2 ⊢ (𝜑 → (( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅 ⊆ 𝑈 ∧ ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))))
52	3, 35, 51	mpbir2and 713	1 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ∩ cin 3899   ⊆ wss 3900   class class class wbr 5089   × cxp 5612   ↾ cres 5616   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343   ↑_m cmap 8745  Basecbs 17112   ⊆_cat cssc 17706  Rngcrng 20063   RngHom crnghm 20345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-ixp 8817  df-ssc 17709  df-ghm 19118  df-abl 19688  df-rng 20064  df-rnghm 20347

This theorem is referenced by:  rnghmsubcsetc  20541