Theorem rnghmsscmap 20706

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 4192	. . 3 ⊢ (Rng ∩ 𝑈) ⊆ 𝑈
3	1, 2	eqsstrdi 3983	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
4		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑎) = (Base‘𝑎)
5		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑏) = (Base‘𝑏)
6	4, 5	rnghmf 20521	. . . . . 6 ⊢ (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
7		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏))
8		fvex 6884	. . . . . . . . . 10 ⊢ (Base‘𝑏) ∈ V
9		fvex 6884	. . . . . . . . . 10 ⊢ (Base‘𝑎) ∈ V
10	8, 9	pm3.2i 475	. . . . . . . . 9 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11		elmapg 8824	. . . . . . . . 9 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏)))
12	10, 11	mp1i 14	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
14	13	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
15	6, 14	syl5 35	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
16	15	ssrdv 3945	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17		ovres 7566	. . . . 5 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
18	17	adantl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
19		eqidd 2766	. . . . 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 6871	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21		fveq2 6871	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
22	20, 21	oveqan12rd 7420	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
23	22	adantl 486	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
24	3	sseld 3938	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝑅 → 𝑎 ∈ 𝑈))
25	24	com12 33	. . . . . . 7 ⊢ (𝑎 ∈ 𝑅 → (𝜑 → 𝑎 ∈ 𝑈))
26	25	adantr 485	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑎 ∈ 𝑈))
27	26	impcom 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑎 ∈ 𝑈)
28	3	sseld 3938	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ 𝑅 → 𝑏 ∈ 𝑈))
29	28	com12 33	. . . . . . 7 ⊢ (𝑏 ∈ 𝑅 → (𝜑 → 𝑏 ∈ 𝑈))
30	29	adantl 486	. . . . . 6 ⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝜑 → 𝑏 ∈ 𝑈))
31	30	impcom 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → 𝑏 ∈ 𝑈)
32		ovexd 7435	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
33	19, 23, 27, 31, 32	ovmpod 7552	. . . 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 3994	. . 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 3208	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36		rnghmfn 20512	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
37	36	a1i 11	. . . 4 ⊢ (𝜑 → RngHom Fn (Rng × Rng))
38		inss1 4191	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
39	1, 38	eqsstrdi 3983	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Rng)
40		xpss12 5667	. . . . 5 ⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
41	39, 39, 40	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42		fnssres 6648	. . . 4 ⊢ (( RngHom Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
43	37, 41, 42	syl2anc 595	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45		ovex 7433	. . . . 5 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
46	44, 45	fnmpoi 8055	. . . 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 3478	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
50	48, 49	syl 18	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
51	43, 47, 50	isssc 17867	. 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 725	1 ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907   class class class wbr 5105   × cxp 5650   ↾ cres 5654   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   ↑_m cmap 8812  Basecbs 17259   ⊆_cat cssc 17854  Rngcrng 20221   RngHom crnghm 20507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-ssc 17857  df-ghm 19275  df-abl 19844  df-rng 20222  df-rnghm 20509

This theorem is referenced by:  rnghmsubcsetc  20709