funcres2c - Metamath Proof Explorer

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Theorem funcres2c 17802

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 865	. . 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 866	. . 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 2731	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		eqid 2731	. . . . . . 7 ⊢ (Hom_f ‘𝐷) = (Hom_f ‘𝐷)
9		funcres2c.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		inss2 4194	. . . . . . . 8 ⊢ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
12	7, 8, 9, 11	fullsubc 17750	. . . . . 6 ⊢ (𝜑 → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
13	12	adantr 481	. . . . 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 17587	. . . . . . 7 ⊢ (Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15		xpss12 5653	. . . . . . . 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 690	. . . . . . 7 ⊢ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17		fnssres 6629	. . . . . . 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 690	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶𝑆)
22	21	ffnd 6674	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
23	21	frnd 6681	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ 𝑆)
24		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
25	5, 7, 24	funcf1 17766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
26	25	frnd 6681	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐸) = (Base‘𝐸)
28		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
29	5, 27, 28	funcf1 17766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
30	29	frnd 6681	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31		funcres2c.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝐷 ↾_s 𝑆)
32	31, 7	ressbasss 17133	. . . . . . . . 9 ⊢ (Base‘𝐸) ⊆ (Base‘𝐷)
33	30, 32	sstrdi 3959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
34	26, 33	jaodan 956	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
35	23, 34	ssind 4197	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36		df-f 6505	. . . . . 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 2731	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
39		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
40		simplrl 775	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥 ∈ 𝐴)
41		simplrr 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦 ∈ 𝐴)
42	5, 6, 38, 39, 40, 41	funcf2 17768	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
43		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
44		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
45		simplrl 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥 ∈ 𝐴)
46		simplrr 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦 ∈ 𝐴)
47	5, 6, 43, 44, 45, 46	funcf2 17768	. . . . . . . . 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 17314	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
51	50	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
52	51	oveqd 7379	. . . . . . . . . 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 6660	. . . . . . . . 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 256	. . . . . . . 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 956	. . . . . . 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 650	. . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
58		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
59	57, 58	ffvelcdmd 7041	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
61	57, 60	ffvelcdmd 7041	. . . . . . . . 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 7526	. . . . . . . 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 4164	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
64	61	elin2d 4164	. . . . . . . . 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 17586	. . . . . . . 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 6660	. . . . . 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 256	. . . . 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 17797	. . . 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 2732	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐶))
71		eqidd 2732	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐶) = (comp_f‘𝐶))
72	7	ressinbas 17140	. . . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
76		eqid 2731	. . . . . . . . . 10 ⊢ (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))
77		eqid 2731	. . . . . . . . . 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 17751	. . . . . . . . 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 495	. . . . . . . 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 481	. . . . . 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 6851	. . . . . . . 8 ⊢ (𝜑 → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
83	78	simprd 496	. . . . . . . 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 481	. . . . . 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 5111	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87		funcrcl 17763	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
88	86, 87	sylbi 216	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
89	88	simpld 495	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
90		df-br 5111	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91		funcrcl 17763	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
92	90, 91	sylbi 216	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
93	92	simpld 495	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → 𝐶 ∈ Cat)
94	89, 93	jaoi 855	. . . . . . . 8 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
95	94	elexd 3466	. . . . . . 7 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
96	95	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
97	31	ovexi 7396	. . . . . . 7 ⊢ 𝐸 ∈ V
98	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99		ovexd 7397	. . . . . 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 17801	. . . . 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 5121	. . . 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 281	. . 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 413	. 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 380	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3446 ∩ cin 3912 ⊆ wss 3913 ⟨cop 4597 class class class wbr 5110 × cxp 5636 ran crn 5639 ↾ cres 5640 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 Basecbs 17094 ↾_s cress 17123 Hom chom 17158 Catccat 17558 Hom_f chomf 17560 comp_fccomf 17561 ↾_cat cresc 17705 Subcatcsubc 17706 Func cfunc 17754

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-hom 17171 df-cco 17172 df-cat 17562 df-cid 17563 df-homf 17564 df-comf 17565 df-ssc 17707 df-resc 17708 df-subc 17709 df-func 17758

This theorem is referenced by: fthres2c 17832 fullres2c 17840

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