funcres2c - Metamath Proof Explorer

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Theorem funcres2c 17617

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 864	. . 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 865	. . 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 2738	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		eqid 2738	. . . . . . 7 ⊢ (Hom_f ‘𝐷) = (Hom_f ‘𝐷)
9		funcres2c.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		inss2 4163	. . . . . . . 8 ⊢ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
12	7, 8, 9, 11	fullsubc 17565	. . . . . 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 17402	. . . . . . 7 ⊢ (Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15		xpss12 5604	. . . . . . . 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 689	. . . . . . 7 ⊢ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17		fnssres 6555	. . . . . . 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 689	. . . . . 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 6601	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
23	21	frnd 6608	. . . . . . 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 17581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
26	25	frnd 6608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27		eqid 2738	. . . . . . . . . . 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 17581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
30	29	frnd 6608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31		funcres2c.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝐷 ↾_s 𝑆)
32	31, 7	ressbasss 16950	. . . . . . . . 9 ⊢ (Base‘𝐸) ⊆ (Base‘𝐷)
33	30, 32	sstrdi 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
34	26, 33	jaodan 955	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
35	23, 34	ssind 4166	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36		df-f 6437	. . . . . 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 2738	. . . . . . . . 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 774	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥 ∈ 𝐴)
41		simplrr 775	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦 ∈ 𝐴)
42	5, 6, 38, 39, 40, 41	funcf2 17583	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
43		eqid 2738	. . . . . . . . . 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 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥 ∈ 𝐴)
46		simplrr 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦 ∈ 𝐴)
47	5, 6, 43, 44, 45, 46	funcf2 17583	. . . . . . . . 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 17129	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
51	50	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
52	51	oveqd 7292	. . . . . . . . . 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 6587	. . . . . . . . 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 955	. . . . . . 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 649	. . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
59	57, 58	ffvelrnd 6962	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60		simprr 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
61	57, 60	ffvelrnd 6962	. . . . . . . . 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 7439	. . . . . . . 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 4133	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
64	61	elin2d 4133	. . . . . . . . 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 17401	. . . . . . . 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 2778	. . . . . . 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 6587	. . . . . 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 17612	. . . 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 2739	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐶))
71		eqidd 2739	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐶) = (comp_f‘𝐶))
72	7	ressinbas 16955	. . . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
75	74	fveq2d 6778	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
76		eqid 2738	. . . . . . . . . 10 ⊢ (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))
77		eqid 2738	. . . . . . . . . 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 17566	. . . . . . . . 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 2778	. . . . . . 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 6778	. . . . . . . 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 2778	. . . . . . 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 5075	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87		funcrcl 17578	. . . . . . . . . . 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 5075	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91		funcrcl 17578	. . . . . . . . . . 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 854	. . . . . . . 8 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
95	94	elexd 3452	. . . . . . 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 7309	. . . . . . 7 ⊢ 𝐸 ∈ V
98	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99		ovexd 7310	. . . . . 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 17616	. . . . 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 5085	. . . 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 381	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 844 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ⟨cop 4567 class class class wbr 5074 × cxp 5587 ran crn 5590 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 ↾_s cress 16941 Hom chom 16973 Catccat 17373 Hom_f chomf 17375 comp_fccomf 17376 ↾_cat cresc 17520 Subcatcsubc 17521 Func cfunc 17569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-hom 16986 df-cco 16987 df-cat 17377 df-cid 17378 df-homf 17379 df-comf 17380 df-ssc 17522 df-resc 17523 df-subc 17524 df-func 17573

This theorem is referenced by: fthres2c 17647 fullres2c 17655

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