funcres2c - Metamath Proof Explorer

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Theorem funcres2c 17996

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 881	. . 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 882	. . 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 2762	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2762	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		eqid 2762	. . . . . . 7 ⊢ (Hom_f ‘𝐷) = (Hom_f ‘𝐷)
9		funcres2c.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		inss2 4186	. . . . . . . 8 ⊢ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
12	7, 8, 9, 11	fullsubc 17943	. . . . . 6 ⊢ (𝜑 → ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
13	12	adantr 486	. . . . 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 17785	. . . . . . 7 ⊢ (Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15		xpss12 5674	. . . . . . . 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 705	. . . . . . 7 ⊢ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17		fnssres 6659	. . . . . . 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 705	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶𝑆)
22	21	ffnd 6707	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
23	21	frnd 6715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ 𝑆)
24		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
25	5, 7, 24	funcf1 17959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
26	25	frnd 6715	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝐸) = (Base‘𝐸)
28		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
29	5, 27, 28	funcf1 17959	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
30	29	frnd 6715	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31		funcres2c.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝐷 ↾_s 𝑆)
32	31, 7	ressbasss 17335	. . . . . . . . 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 972	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
35	23, 34	ssind 4189	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36		df-f 6541	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷))))
37	22, 35, 36	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
38		eqid 2762	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
39		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
40		simplrl 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥 ∈ 𝐴)
41		simplrr 790	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦 ∈ 𝐴)
42	5, 6, 38, 39, 40, 41	funcf2 17961	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
43		eqid 2762	. . . . . . . . . 10 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
44		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
45		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥 ∈ 𝐴)
46		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦 ∈ 𝐴)
47	5, 6, 43, 44, 45, 46	funcf2 17961	. . . . . . . . 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 17507	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
50	48, 49	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
51	50	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
52	51	oveqd 7433	. . . . . . . . . 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 6691	. . . . . . . . 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 260	. . . . . . . 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 972	. . . . . . 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 665	. . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
58		simprl 783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
59	57, 58	ffvelcdmd 7081	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60		simprr 785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
61	57, 60	ffvelcdmd 7081	. . . . . . . . 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 7583	. . . . . . . 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 4154	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
64	61	elin2d 4154	. . . . . . . . 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 17784	. . . . . . . 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 2797	. . . . . . 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 6691	. . . . . 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 260	. . . . 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 17990	. . . 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 2763	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐶))
71		eqidd 2763	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐶) = (comp_f‘𝐶))
72	7	ressinbas 17341	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑉 → (𝐷 ↾_s 𝑆) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
73	48, 72	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ↾_s 𝑆) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
74	31, 73	eqtrid 2809	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
75	74	fveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
76		eqid 2762	. . . . . . . . . 10 ⊢ (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))
77		eqid 2762	. . . . . . . . . 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 17944	. . . . . . . . 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 500	. . . . . . . 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 2797	. . . . . . 7 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
81	80	adantr 486	. . . . . 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 6886	. . . . . . . 8 ⊢ (𝜑 → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
83	78	simprd 501	. . . . . . . 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 2797	. . . . . . 7 ⊢ (𝜑 → (comp_f‘𝐸) = (comp_f‘(𝐷 ↾_cat ((Hom_f ‘𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
85	84	adantr 486	. . . . . 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 5108	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87		funcrcl 17956	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
88	86, 87	sylbi 220	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
89	88	simpld 500	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
90		df-br 5108	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91		funcrcl 17956	. . . . . . . . . . 11 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
92	90, 91	sylbi 220	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
93	92	simpld 500	. . . . . . . . 9 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 → 𝐶 ∈ Cat)
94	89, 93	jaoi 871	. . . . . . . 8 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
95	94	elexd 3476	. . . . . . 7 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
96	95	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
97	31	ovexi 7450	. . . . . . 7 ⊢ 𝐸 ∈ V
98	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99		ovexd 7451	. . . . . 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 17995	. . . . 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 5118	. . . 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 285	. . 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 418	. 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 382	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ∩ cin 3901 ⊆ wss 3902 ⟨cop 4593 class class class wbr 5107 × cxp 5657 ran crn 5660 ↾ cres 5661 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 Basecbs 17305 ↾_s cress 17326 Hom chom 17357 Catccat 17756 Hom_f chomf 17758 comp_fccomf 17759 ↾_cat cresc 17901 Subcatcsubc 17902 Func cfunc 17947

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 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 proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-hom 17370 df-cco 17371 df-cat 17760 df-cid 17761 df-homf 17762 df-comf 17763 df-ssc 17903 df-resc 17904 df-subc 17905 df-func 17951

This theorem is used by: fthres2c 18026 fullres2c 18034

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