Theorem rnghmsscmap 20567

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 4191	. . 3 ⊢ (Rng ∩ 𝑈) ⊆ 𝑈
3	1, 2	eqsstrdi 3979	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
4		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
5		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
6	4, 5	rnghmf 20388	. . . . . 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 6848	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
9		fvex 6848	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
10	8, 9	pm3.2i 470	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11		elmapg 8780	. . . . . . . . 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 3940	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17		ovres 7526	. . . . 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 2738	. . . . 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 6835	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21		fveq2 6835	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
22	20, 21	oveqan12rd 7380	. . . . . 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 3933	. . . . . . . 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 3933	. . . . . . . 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 7395	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
33	19, 23, 27, 31, 32	ovmpod 7512	. . . 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 3990	. . 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 3180	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36		rnghmfn 20379	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
38		inss1 4190	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
39	1, 38	eqsstrdi 3979	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Rng)
40		xpss12 5640	. . . . 5 ⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
41	39, 39, 40	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42		fnssres 6616	. . . 4 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
43	37, 41, 42	syl2anc 585	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45		ovex 7393	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
46	44, 45	fnmpoi 8016	. . . 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 3462	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
51	43, 47, 50	isssc 17748	. 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 714	1 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5099   × cxp 5623   ↾ cres 5627   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362   ↑_m cmap 8767  Basecbs 17140   ⊆_cat cssc 17735  Rngcrng 20091   RngHom crnghm 20374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-ixp 8840  df-ssc 17738  df-ghm 19146  df-abl 19716  df-rng 20092  df-rnghm 20376

This theorem is referenced by:  rnghmsubcsetc  20570