Theorem islmd 49483

Description: The universal property of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
islmd.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
islmd.a	⊢ 𝐴 = (Base‘𝐶)
islmd.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
islmd.b	⊢ 𝐵 = (Base‘𝐷)
islmd.h	⊢ 𝐻 = (Hom ‘𝐶)
islmd.x	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
islmd	⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))

Distinct variable groups:   · ,𝑗   𝐴,𝑎,𝑗,𝑚,𝑥   𝐵,𝑗   𝐶,𝑎,𝑗,𝑚,𝑥   𝐷,𝑎,𝑗,𝑚,𝑥   𝐹,𝑎,𝑗,𝑚,𝑥   𝑗,𝐻,𝑚   𝐿,𝑎,𝑗,𝑚,𝑥   𝑁,𝑎,𝑗,𝑚,𝑥   𝑅,𝑎,𝑗,𝑚,𝑥   𝑋,𝑎,𝑗,𝑚,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑚,𝑎)   · (𝑥,𝑚,𝑎)   𝐻(𝑥,𝑎)

Proof of Theorem islmd

Step	Hyp	Ref	Expression
1		lmdfval2 49475	. . . 4 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
2		islmd.l	. . . . . 6 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3	2	fveq2i 6878	. . . . 5 ⊢ (oppFunc‘𝐿) = (oppFunc‘(𝐶Δfunc𝐷))
4	3	oveq1i 7413	. . . 4 ⊢ ((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
5	1, 4	eqtr4i 2761	. . 3 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = ((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
6	5	breqi 5125	. 2 ⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑅 ↔ 𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅)
7		id 22	. . . . . 6 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅)
8	7	up1st2nd 49067	. . . . 5 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝑋(⟨(1st ‘(oppFunc‘𝐿)), (2nd ‘(oppFunc‘𝐿))⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅)
9		eqid 2735	. . . . 5 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
10		islmd.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
11	8, 9, 10	oppcuprcl4 49080	. . . 4 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝑋 ∈ 𝐴)
12		eqid 2735	. . . . . . . 8 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
13		eqid 2735	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
14	13	fucbas 17974	. . . . . . . 8 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
15	8, 12, 14	oppcuprcl3 49081	. . . . . . 7 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝐹 ∈ (𝐷 Func 𝐶))
16		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
17	16	func1st2nd 48991	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
18	17	funcrcl3 48993	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
19	17	funcrcl2 48992	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐷 ∈ Cat)
20	2, 18, 19, 13	diagcl 18251	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
21		oppfval2 49031	. . . . . . . . . 10 ⊢ (𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)) → (oppFunc‘𝐿) = ⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (oppFunc‘𝐿) = ⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩)
23	22	oveq1d 7418	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → ((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = (⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹))
24	23	breqd 5130	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ 𝑋(⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅))
25	11, 15, 24	syl2anc 584	. . . . . 6 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ 𝑋(⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅))
26	25	ibi 267	. . . . 5 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝑋(⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅)
27		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
28	13, 27	fuchom 17975	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
29	26, 12, 28	oppcuprcl5 49082	. . . 4 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹))
30	11, 29	jca 511	. . 3 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)))
31	27	natrcl 17964	. . . . . 6 ⊢ (𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹) → (((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶) ∧ 𝐹 ∈ (𝐷 Func 𝐶)))
32	31	simprd 495	. . . . 5 ⊢ (𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹) → 𝐹 ∈ (𝐷 Func 𝐶))
33	32, 24	sylan2 593	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ 𝑋(⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅))
34		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
35		eqid 2735	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
36	32	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝐹 ∈ (𝐷 Func 𝐶))
37	32, 20	sylan2 593	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
38	37	func1st2nd 48991	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
39		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝑋 ∈ 𝐴)
40		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹))
41	10, 14, 34, 28, 35, 36, 38, 39, 40, 9, 12	oppcup 49088	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → (𝑋(⟨(1st ‘𝐿), tpos (2nd ‘𝐿)⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚))))
42		islmd.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐷)
43	32, 18	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝐶 ∈ Cat)
44	43	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝐶 ∈ Cat)
45	32, 19	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → 𝐷 ∈ Cat)
46	45	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝐷 ∈ Cat)
47		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝑥 ∈ 𝐴)
48	39	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝑋 ∈ 𝐴)
49		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝑚 ∈ (𝑥𝐻𝑋))
50	2, 10, 42, 34, 44, 46, 47, 48, 49	diag2 18255	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → ((𝑥(2nd ‘𝐿)𝑋)‘𝑚) = (𝐵 × {𝑚}))
51	50	oveq2d 7419	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚)) = (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)(𝐵 × {𝑚})))
52		islmd.x	. . . . . . . . 9 ⊢ · = (comp‘𝐶)
53	2, 10, 42, 34, 44, 46, 47, 48, 49, 27	diag2cl 18256	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝐵 × {𝑚}) ∈ (((1st ‘𝐿)‘𝑥)𝑁((1st ‘𝐿)‘𝑋)))
54	40	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹))
55	13, 27, 42, 52, 35, 53, 54	fucco 17976	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)(𝐵 × {𝑚})) = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘𝐹)‘𝑗))((𝐵 × {𝑚})‘𝑗))))
56	44	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
57	46	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
58	47	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
59		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
60		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ 𝐵)
61	2, 56, 57, 10, 58, 59, 42, 60	diag11 18253	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗) = 𝑥)
62	48	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
63		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
64	2, 56, 57, 10, 62, 63, 42, 60	diag11 18253	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗) = 𝑋)
65	61, 64	opeq12d 4857	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → ⟨((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ = ⟨𝑥, 𝑋⟩)
66	65	oveq1d 7418	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → (⟨((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘𝐹)‘𝑗)) = (⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗)))
67		eqidd 2736	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → (𝑅‘𝑗) = (𝑅‘𝑗))
68		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
69	68	fvconst2 7195	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
70	69	adantl 481	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
71	66, 67, 70	oveq123d 7424	. . . . . . . . 9 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) ∧ 𝑗 ∈ 𝐵) → ((𝑅‘𝑗)(⟨((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘𝐹)‘𝑗))((𝐵 × {𝑚})‘𝑗)) = ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))
72	71	mpteq2dva 5214	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘𝐹)‘𝑗))((𝐵 × {𝑚})‘𝑗))) = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚)))
73	51, 55, 72	3eqtrd 2774	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚)) = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚)))
74	73	eqeq2d 2746	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) ∧ 𝑚 ∈ (𝑥𝐻𝑋)) → (𝑎 = (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚)) ↔ 𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
75	74	reubidva 3375	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹))) → (∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚)) ↔ ∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
76	75	2ralbidva 3203	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → (∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑅(⟨((1st ‘𝐿)‘𝑥), ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))𝐹)((𝑥(2nd ‘𝐿)𝑋)‘𝑚)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
77	33, 41, 76	3bitrd 305	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) → (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
78	30, 77	biadanii 821	. 2 ⊢ (𝑋((oppFunc‘𝐿)((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
79	6, 78	bitri 275	1 ⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  {csn 4601  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8222  Basecbs 17226  Hom chom 17280  compcco 17281  Catccat 17674  oppCatcoppc 17721   Func cfunc 17865   Nat cnat 17955   FuncCat cfuc 17956  Δ_funccdiag 18222  oppFunccoppf 49019   UP cup 49056   Limit clmd 49465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17293  df-cco 17294  df-cat 17678  df-cid 17679  df-oppc 17722  df-func 17869  df-nat 17957  df-fuc 17958  df-xpc 18182  df-1stf 18183  df-curf 18224  df-diag 18226  df-oppf 49020  df-up 49057  df-lmd 49467

This theorem is referenced by: (None)