funcres2c - Metamath Proof Explorer

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Theorem funcres2c 17827

Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)

**Hypotheses**
Ref	Expression
funcres2c.a	⊢ 𝐴 = (Base‘𝐶)
funcres2c.e	⊢ 𝐸 = (𝐷 ↾_s 𝑆)
funcres2c.d	⊢ (𝜑 → 𝐷 ∈ Cat)
funcres2c.r	⊢ (𝜑 → 𝑆 ∈ 𝑉)
funcres2c.1	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)

**Assertion**
Ref	Expression
funcres2c	⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))

Proof of Theorem funcres2c Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 867	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺))
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)))
3		olc 868	. . 3 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺))
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐸)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)))
5		funcres2c.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2736	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		eqid 2736	. . . . . . 7 ⊢ (Hom_f ‘𝐷) = (Hom_f ‘𝐷)
9		funcres2c.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		inss2 4190	. . . . . . . 8 ⊢ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
12	7, 8, 9, 11	fullsubc 17774	. . . . . 6 ⊢ (𝜑 → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
14	8, 7	homffn 17616	. . . . . . 7 ⊢ (Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15		xpss12 5639	. . . . . . . 8 ⊢ (((𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷) ∧ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)) → ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
16	10, 10, 15	mp2an 692	. . . . . . 7 ⊢ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17		fnssres 6615	. . . . . . 7 ⊢ (((Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) ∧ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))) → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))
18	14, 16, 17	mp2an 692	. . . . . 6 ⊢ ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))
19	18	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))
20		funcres2c.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶𝑆)
22	21	ffnd 6663	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
23	21	frnd 6670	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ 𝑆)
24		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
25	5, 7, 24	funcf1 17790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
26	25	frnd 6670	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐸) = (Base‘𝐸)
28		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
29	5, 27, 28	funcf1 17790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
30	29	frnd 6670	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31		funcres2c.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝐷 ↾_s 𝑆)
32	31, 7	ressbasss 17166	. . . . . . . . 9 ⊢ (Base‘𝐸) ⊆ (Base‘𝐷)
33	30, 32	sstrdi 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
34	26, 33	jaodan 959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
35	23, 34	ssind 4193	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36		df-f 6496	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷))))
37	22, 35, 36	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
38		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
39		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
40		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥 ∈ 𝐴)
41		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦 ∈ 𝐴)
42	5, 6, 38, 39, 40, 41	funcf2 17792	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
43		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
44		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
45		simplrl 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥 ∈ 𝐴)
46		simplrr 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦 ∈ 𝐴)
47	5, 6, 43, 44, 45, 46	funcf2 17792	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
48		funcres2c.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
49	31, 38	resshom 17338	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
51	50	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
52	51	oveqd 7375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
53	52	feq3d 6647	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦))))
54	47, 53	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
55	42, 54	jaodan 959	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
56	55	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
57	37	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
58		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
59	57, 58	ffvelcdmd 7030	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
61	57, 60	ffvelcdmd 7030	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ (𝑆 ∩ (Base‘𝐷)))
62	59, 61	ovresd 7525	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom_f ‘𝐷)(𝐹‘𝑦)))
63	59	elin2d 4157	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
64	61	elin2d 4157	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ (Base‘𝐷))
65	8, 7, 38, 63, 64	homfval 17615	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)(Hom_f ‘𝐷)(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
66	62, 65	eqtrd 2771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
67	66	feq3d 6647	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
68	56, 67	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹‘𝑦)))
69	5, 6, 13, 19, 37, 68	funcres2b 17821	. . . 4 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))𝐺))
70		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐶))
71		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐶) = (comp_f‘𝐶))
72	7	ressinbas 17172	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑉 → (𝐷 ↾_s 𝑆) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
73	48, 72	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ↾_s 𝑆) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
74	31, 73	eqtrid 2783	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
75	74	fveq2d 6838	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
76		eqid 2736	. . . . . . . . . 10 ⊢ (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))
77		eqid 2736	. . . . . . . . . 10 ⊢ (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) = (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))
78	7, 8, 9, 11, 76, 77	fullresc 17775	. . . . . . . . 9 ⊢ (𝜑 → ((Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))) = (Hom_f ‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))) ∧ (comp_f‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))) = (comp_f‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))))
79	78	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))) = (Hom_f ‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
80	75, 79	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
81	80	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
82	74	fveq2d 6838	. . . . . . . 8 ⊢ (𝜑 → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
83	78	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (comp_f‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))) = (comp_f‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
84	82, 83	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
85	84	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
86		df-br 5099	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87		funcrcl 17787	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
88	86, 87	sylbi 217	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
89	88	simpld 494	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
90		df-br 5099	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91		funcrcl 17787	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
92	90, 91	sylbi 217	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
93	92	simpld 494	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → 𝐶 ∈ Cat)
94	89, 93	jaoi 857	. . . . . . . 8 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
95	94	elexd 3464	. . . . . . 7 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
96	95	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
97	31	ovexi 7392	. . . . . . 7 ⊢ 𝐸 ∈ V
98	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99		ovexd 7393	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) ∈ V)
100	70, 71, 81, 85, 96, 96, 98, 99	funcpropd 17826	. . . . 5 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝐶 Func 𝐸) = (𝐶 Func (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
101	100	breqd 5109	. . . 4 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐸)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))𝐺))
102	69, 101	bitr4d 282	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))
103	102	ex 412	. 2 ⊢ (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺)))
104	2, 4, 103	pm5.21ndd 379	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ∩ cin 3900 ⊆ wss 3901 ⟨cop 4586 class class class wbr 5098 × cxp 5622 ran crn 5625 ↾ cres 5626 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 ↾_s cress 17157 Hom chom 17188 Catccat 17587 Hom_f chomf 17589 comp_fccomf 17590 ↾_cat cresc 17732 Subcatcsubc 17733 Func cfunc 17778

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-hom 17201 df-cco 17202 df-cat 17591 df-cid 17592 df-homf 17593 df-comf 17594 df-ssc 17734 df-resc 17735 df-subc 17736 df-func 17782

This theorem is referenced by: fthres2c 17857 fullres2c 17865

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