Theorem funcres2b 17947

Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
funcres2b.a	⊢ 𝐴 = (Base‘𝐶)
funcres2b.h	⊢ 𝐻 = (Hom ‘𝐶)
funcres2b.r	⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷))
funcres2b.s	⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆))
funcres2b.1	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
funcres2b.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))

Assertion

Ref	Expression
funcres2b	⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem funcres2b

Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5148	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2		funcrcl 17913	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	sylbi 217	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simpld 494	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
5	4	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat))
6		df-br 5148	. . . . 5 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)))
7		funcrcl 17913	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
8	6, 7	sylbi 217	. . . 4 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
9	8	simpld 494	. . 3 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 → 𝐶 ∈ Cat)
10	9	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 → 𝐶 ∈ Cat))
11		funcres2b.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
12		funcres2b.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷))
13		funcres2b.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆))
14		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	12, 13, 14	subcss1 17892	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
16	11, 15	fssd 6753	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐷))
17		eqid 2734	. . . . . . . . . 10 ⊢ (𝐷 ↾cat 𝑅) = (𝐷 ↾cat 𝑅)
18		subcrcl 17863	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
19	12, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	17, 14, 19, 13, 15	rescbas 17876	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘(𝐷 ↾cat 𝑅)))
21	20	feq3d 6723	. . . . . . . 8 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
22	11, 21	mpbid 232	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅)))
23	16, 22	2thd 265	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
25		funcres2b.2	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
26	25	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
27	26	frnd 6744	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
28	12	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
29	13	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	11	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶𝑆)
32		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
33	31, 32	ffvelcdmd 7104	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ 𝑆)
34		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
35	31, 34	ffvelcdmd 7104	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝑆)
36	28, 29, 30, 33, 35	subcss2 17893	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
37	27, 36	sstrd 4005	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
38	37, 27	2thd 265	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦))))
39	38	anbi2d 630	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦)))))
40		df-f 6566	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
41		df-f 6566	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦))))
42	39, 40, 41	3bitr4g 314	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦))))
43	17, 14, 19, 13, 15	reschom 17878	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
44	43	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
45	44	oveqd 7447	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
46	45	feq3d 6723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦))))
47	42, 46	bitrd 279	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦))))
48	47	ralrimivva 3199	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦))))
49		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50		df-ov 7433	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
51	49, 50	eqtr4di 2792	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
52		vex 3481	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53		vex 3481	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
54	52, 53	op1std 8022	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
55	54	fveq2d 6910	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
56	52, 53	op2ndd 8023	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
57	56	fveq2d 6910	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
58	55, 57	oveq12d 7448	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
59		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60		df-ov 7433	. . . . . . . . . . . . . . 15 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
61	59, 60	eqtr4di 2792	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
62	58, 61	oveq12d 7448	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
63	51, 62	eleq12d 2832	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦))))
64		ovex 7463	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ∈ V
65		ovex 7463	. . . . . . . . . . . . 13 ⊢ (𝑥𝐻𝑦) ∈ V
66	64, 65	elmap 8909	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
67	63, 66	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
68	55, 57	oveq12d 7448	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
69	68, 61	oveq12d 7448	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
70	51, 69	eleq12d 2832	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦))))
71		ovex 7463	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ∈ V
72	71, 65	elmap 8909	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
73	70, 72	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦))))
74	67, 73	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))))
75	74	ralxp 5854	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝐴 × 𝐴)((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦))))
76	48, 75	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → ∀𝑧 ∈ (𝐴 × 𝐴)((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
77		ralbi 3100	. . . . . . . 8 ⊢ (∀𝑧 ∈ (𝐴 × 𝐴)((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) → (∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
78	76, 77	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
79	78	3anbi3d 1441	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))))
80		elixp2 8939	. . . . . 6 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
81		elixp2 8939	. . . . . 6 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
82	79, 80, 81	3bitr4g 314	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ 𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
83	12	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (Subcat‘𝐷))
84	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85		eqid 2734	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐹:𝐴⟶𝑆)
87	86	ffvelcdmda 7103	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑆)
88	17, 83, 84, 85, 87	subcid 17897	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → ((Id‘𝐷)‘(𝐹‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)))
89	88	eqeq2d 2745	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ↔ ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥))))
90		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐷) = (comp‘𝐷)
91	17, 14, 19, 13, 15, 90	rescco 17880	. . . . . . . . . . . . 13 ⊢ (𝜑 → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
92	91	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
93	92	oveqd 7447	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧)) = (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧)))
94	93	oveqd 7447	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))
95	94	eqeq2d 2745	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))
96	95	2ralbidv 3218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))
97	96	2ralbidv 3218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))
98	89, 97	anbi12d 632	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → ((((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))) ↔ (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))))
99	98	ralbidva 3173	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))) ↔ ∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))))
100	24, 82, 99	3anbi123d 1435	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → ((𝐹:𝐴⟶(Base‘𝐷) ∧ 𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))) ↔ (𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅)) ∧ 𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))))
101		funcres2b.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
102		funcres2b.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
103		eqid 2734	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
104		eqid 2734	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
105		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
106	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
107	101, 14, 102, 30, 103, 85, 104, 90, 105, 106	isfunc 17914	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹:𝐴⟶(Base‘𝐷) ∧ 𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))))
108		eqid 2734	. . . . 5 ⊢ (Base‘(𝐷 ↾cat 𝑅)) = (Base‘(𝐷 ↾cat 𝑅))
109		eqid 2734	. . . . 5 ⊢ (Hom ‘(𝐷 ↾cat 𝑅)) = (Hom ‘(𝐷 ↾cat 𝑅))
110		eqid 2734	. . . . 5 ⊢ (Id‘(𝐷 ↾cat 𝑅)) = (Id‘(𝐷 ↾cat 𝑅))
111		eqid 2734	. . . . 5 ⊢ (comp‘(𝐷 ↾cat 𝑅)) = (comp‘(𝐷 ↾cat 𝑅))
112	17, 12	subccat 17898	. . . . . 6 ⊢ (𝜑 → (𝐷 ↾cat 𝑅) ∈ Cat)
113	112	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐷 ↾cat 𝑅) ∈ Cat)
114	101, 108, 102, 109, 103, 110, 104, 111, 105, 113	isfunc 17914	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ↔ (𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅)) ∧ 𝐺 ∈ X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st ‘𝑧))(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘(𝐷 ↾cat 𝑅))(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))))))
115	100, 107, 114	3bitr4d 311	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))
116	115	ex 412	. 2 ⊢ (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺)))
117	5, 10, 116	pm5.21ndd 379	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ⊆ wss 3962  ⟨cop 4636   class class class wbr 5147   × cxp 5686  ran crn 5689   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011   ↑_m cmap 8864  Xcixp 8935  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Idccid 17709   ↾_cat cresc 17855  Subcatcsubc 17856   Func cfunc 17904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-homf 17714  df-ssc 17857  df-resc 17858  df-subc 17859  df-func 17908

This theorem is referenced by:  funcres2  17948  funcres2c  17954  fthres2b  17983