funcres2c - Metamath Proof Explorer

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Theorem funcres2c 18107

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 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		eqid 2763	. . . . . . 7 ⊢ (Hom_f ‘𝐷) = (Hom_f ‘𝐷)
9		funcres2c.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		inss2 4137	. . . . . . . 8 ⊢ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
12	7, 8, 9, 11	fullsubc 18054	. . . . . 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 17896	. . . . . . 7 ⊢ (Hom_f ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15		xpss12 5670	. . . . . . . 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 6684	. . . . . . 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 6732	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
23	21	frnd 6740	. . . . . . 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 18070	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
26	25	frnd 6740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27		eqid 2763	. . . . . . . . . . 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 18070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
30	29	frnd 6740	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31		funcres2c.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝐷 ↾_s 𝑆)
32	31, 7	ressbasss 17446	. . . . . . . . 9 ⊢ (Base‘𝐸) ⊆ (Base‘𝐷)
33	30, 32	sstrdi 3925	. . . . . . . 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 4140	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36		df-f 6562	. . . . . 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 2763	. . . . . . . . 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 18072	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
43		eqid 2763	. . . . . . . . . 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 18072	. . . . . . . . 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 17618	. . . . . . . . . . . . 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 7460	. . . . . . . . . 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 6716	. . . . . . . . 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 7109	. . . . . . . . 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 7109	. . . . . . . . 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 7611	. . . . . . . 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 4105	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
64	61	elin2d 4105	. . . . . . . . 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 17895	. . . . . . . 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 2798	. . . . . . 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 6716	. . . . . 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 18101	. . . 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 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐶))
71		eqidd 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → (comp_f‘𝐶) = (comp_f‘𝐶))
72	7	ressinbas 17452	. . . . . . . . . . 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 2810	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))))
75	74	fveq2d 6912	. . . . . . . 8 ⊢ (𝜑 → (Hom_f ‘𝐸) = (Hom_f ‘(𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))))
76		eqid 2763	. . . . . . . . . 10 ⊢ (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷))) = (𝐷 ↾_s (𝑆 ∩ (Base‘𝐷)))
77		eqid 2763	. . . . . . . . . 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 18055	. . . . . . . . 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 2798	. . . . . . 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 6912	. . . . . . . 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 2798	. . . . . . 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 5107	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87		funcrcl 18067	. . . . . . . . . . 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 5107	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91		funcrcl 18067	. . . . . . . . . . 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 3477	. . . . . . 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 7477	. . . . . . 7 ⊢ 𝐸 ∈ V
98	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺 ∨ 𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99		ovexd 7478	. . . . . 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 18106	. . . . 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 5117	. . . 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 2147 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ⟨cop 4838 class class class wbr 5106 × cxp 5653 ran crn 5656 ↾ cres 5657 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 ↾_s cress 17437 Hom chom 17468 Catccat 17867 Hom_f chomf 17869 comp_fccomf 17870 ↾_cat cresc 18012 Subcatcsubc 18013 Func cfunc 18058

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-hom 17481 df-cco 17482 df-cat 17871 df-cid 17872 df-homf 17873 df-comf 17874 df-ssc 18014 df-resc 18015 df-subc 18016 df-func 18062

This theorem is used by: fthres2c 18137 fullres2c 18145

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