Theorem rnghmsscmap2 20739

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.)

Hypotheses

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

Assertion

Ref	Expression
rnghmsscmap2	⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Distinct variable group:   𝑥,𝑅,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑈(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem rnghmsscmap2

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

Step	Hyp	Ref	Expression
1		ssidd 3959	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑅)
2		eqid 2762	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
3		eqid 2762	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
4	2, 3	rnghmf 20537	. . . . . 6 ⊢ (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
5		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
6		fvex 6894	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
7		fvex 6894	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
8	6, 7	pm3.2i 475	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9		elmapg 8834	. . . . . . . . 9 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
10	8, 9	mp1i 14	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
11	5, 10	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
12	11	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
13	4, 12	syl5 35	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
14	13	ssrdv 3942	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
15		ovres 7578	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
16	15	adantl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
17		eqidd 2763	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
18		fveq2 6881	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
20	18, 19	oveqan12rd 7432	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
21	20	adantl 486	. . . . . 6 ⊢ (((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
22		simpl 487	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
23		simpr 489	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
24		ovexd 7447	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
25	17, 21, 22, 23, 24	ovmpod 7564	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
26	25	adantl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
27	14, 16, 26	3sstr4d 3991	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
28	27	ralrimivva 3207	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
29		rnghmfn 20528	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
31		rnghmsscmap.r	. . . . . 6 ⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈))
32		inss1 4188	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
33	31, 32	eqsstrdi 3980	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Rng)
34		xpss12 5675	. . . . 5 ⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
35	33, 33, 34	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
36		fnssres 6658	. . . 4 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
37	30, 35, 36	syl2anc 595	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38		eqid 2762	. . . . 5 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
39		ovex 7445	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
40	38, 39	fnmpoi 8065	. . . 4 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅)
41	40	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅))
42		incom 4161	. . . . 5 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
43		rnghmsscmap.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
44		inex1g 5287	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
45	43, 44	syl 18	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
46	42, 45	eqeltrid 2866	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
47	31, 46	eqeltrd 2862	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
48	37, 41, 47	isssc 17883	. 2 ⊢ (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	1, 28, 48	mpbir2and 725	1 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5108   × cxp 5658   ↾ cres 5662   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8822  Basecbs 17275   ⊆_cat cssc 17870  Rngcrng 20236   RngHom crnghm 20523

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-map 8824  df-ixp 8894  df-ssc 17873  df-ghm 19290  df-abl 19859  df-rng 20237  df-rnghm 20525

This theorem is used by: (None)