Theorem funcres2b 17906

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 5095	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2		funcrcl 17872	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	sylbi 219	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simpld 497	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
5	4	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat))
6		df-br 5095	. . . . 5 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)))
7		funcrcl 17872	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
8	6, 7	sylbi 219	. . . 4 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
9	8	simpld 497	. . 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 2756	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	12, 13, 14	subcss1 17851	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
16	11, 15	fssd 6698	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐷))
17		eqid 2756	. . . . . . . . . 10 ⊢ (𝐷 ↾cat 𝑅) = (𝐷 ↾cat 𝑅)
18		subcrcl 17825	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
19	12, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	17, 14, 19, 13, 15	rescbas 17838	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘(𝐷 ↾cat 𝑅)))
21	20	feq3d 6665	. . . . . . . 8 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
22	11, 21	mpbid 234	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅)))
23	16, 22	2thd 267	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
24	23	adantr 483	. . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
27	26	frnd 6689	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
28	12	ad2antrr 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
29	13	ad2antrr 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30		eqid 2756	. . . . . . . . . . . . . . . 16 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	11	ad2antrr 734	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶𝑆)
32		simprl 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
33	31, 32	ffvelcdmd 7055	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ 𝑆)
34		simprr 780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
35	31, 34	ffvelcdmd 7055	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝑆)
36	28, 29, 30, 33, 35	subcss2 17852	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
37	27, 36	sstrd 3941	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
38	37, 27	2thd 267	. . . . . . . . . . . . 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 638	. . . . . . . . . . . 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 6514	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
41		df-f 6514	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦))))
42	39, 40, 41	3bitr4g 316	. . . . . . . . . . 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 17839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
44	43	ad2antrr 734	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
45	44	oveqd 7402	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
46	45	feq3d 6665	. . . . . . . . . . 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 281	. . . . . . . . . 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 6856	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50		df-ov 7388	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
51	49, 50	eqtr4di 2809	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
52		vex 3452	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53		vex 3452	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
54	52, 53	op1std 7969	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
55	54	fveq2d 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
56	52, 53	op2ndd 7970	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
57	56	fveq2d 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
58	55, 57	oveq12d 7403	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
59		fveq2 6856	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60		df-ov 7388	. . . . . . . . . . . . . . 15 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
61	59, 60	eqtr4di 2809	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
62	58, 61	oveq12d 7403	. . . . . . . . . . . . 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 2850	. . . . . . . . . . . 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 7418	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ∈ V
65		ovex 7418	. . . . . . . . . . . . 13 ⊢ (𝑥𝐻𝑦) ∈ V
66	64, 65	elmap 8842	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
67	63, 66	bitrdi 289	. . . . . . . . . . 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 7403	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . 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 2850	. . . . . . . . . . . 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 7418	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ∈ V
72	71, 65	elmap 8842	. . . . . . . . . . . 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 289	. . . . . . . . . . 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 347	. . . . . . . . . 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 5806	. . . . . . . . 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 236	. . . . . . . 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 3111	. . . . . . . 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 1457	. . . . . 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 8872	. . . . . 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 8872	. . . . . 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 316	. . . . 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 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (Subcat‘𝐷))
84	13	ad2antrr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85		eqid 2756	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	11	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐹:𝐴⟶𝑆)
87	86	ffvelcdmda 7054	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑆)
88	17, 83, 84, 85, 87	subcid 17856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → ((Id‘𝐷)‘(𝐹‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)))
89	88	eqeq2d 2767	. . . . . . 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 2756	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐷) = (comp‘𝐷)
91	17, 14, 19, 13, 15, 90	rescco 17841	. . . . . . . . . . . . 13 ⊢ (𝜑 → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
92	91	ad2antrr 734	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
93	92	oveqd 7402	. . . . . . . . . . 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 7402	. . . . . . . . . 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 2767	. . . . . . . . 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 3220	. . . . . . . 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 3220	. . . . . . 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 640	. . . . . 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 3177	. . . . 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 1451	. . . 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 2756	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
104		eqid 2756	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
105		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
106	19	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
107	101, 14, 102, 30, 103, 85, 104, 90, 105, 106	isfunc 17873	. . . 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 2756	. . . . 5 ⊢ (Base‘(𝐷 ↾cat 𝑅)) = (Base‘(𝐷 ↾cat 𝑅))
109		eqid 2756	. . . . 5 ⊢ (Hom ‘(𝐷 ↾cat 𝑅)) = (Hom ‘(𝐷 ↾cat 𝑅))
110		eqid 2756	. . . . 5 ⊢ (Id‘(𝐷 ↾cat 𝑅)) = (Id‘(𝐷 ↾cat 𝑅))
111		eqid 2756	. . . . 5 ⊢ (comp‘(𝐷 ↾cat 𝑅)) = (comp‘(𝐷 ↾cat 𝑅))
112	17, 12	subccat 17857	. . . . . 6 ⊢ (𝜑 → (𝐷 ↾cat 𝑅) ∈ Cat)
113	112	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐷 ↾cat 𝑅) ∈ Cat)
114	101, 108, 102, 109, 103, 110, 104, 111, 105, 113	isfunc 17873	. . . 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 313	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))
116	115	ex 415	. 2 ⊢ (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺)))
117	5, 10, 116	pm5.21ndd 381	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∀wral 3070  Vcvv 3448   ⊆ wss 3899  ⟨cop 4582   class class class wbr 5094   × cxp 5638  ran crn 5641   Fn wfn 6505  ⟶wf 6506  ‘cfv 6510  (class class class)co 7385  1^st c1st 7957  2^nd c2nd 7958   ↑_m cmap 8796  Xcixp 8868  Basecbs 17221  Hom chom 17273  compcco 17274  Catccat 17672  Idccid 17673   ↾_cat cresc 17817  Subcatcsubc 17818   Func cfunc 17863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-cnex 11119  ax-resscn 11120  ax-1cn 11121  ax-icn 11122  ax-addcl 11123  ax-addrcl 11124  ax-mulcl 11125  ax-mulrcl 11126  ax-mulcom 11127  ax-addass 11128  ax-mulass 11129  ax-distr 11130  ax-i2m1 11131  ax-1ne0 11132  ax-1rid 11133  ax-rnegex 11134  ax-rrecex 11135  ax-cnre 11136  ax-pre-lttri 11137  ax-pre-lttrn 11138  ax-pre-ltadd 11139  ax-pre-mulgt0 11140

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-nel 3056  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-er 8666  df-map 8798  df-pm 8799  df-ixp 8869  df-en 8917  df-dom 8918  df-sdom 8919  df-pnf 11208  df-mnf 11209  df-xr 11210  df-ltxr 11211  df-le 11212  df-sub 11406  df-neg 11407  df-nn 12201  df-2 12270  df-3 12271  df-4 12272  df-5 12273  df-6 12274  df-7 12275  df-8 12276  df-9 12277  df-n0 12472  df-z 12559  df-dec 12679  df-sets 17176  df-slot 17194  df-ndx 17206  df-base 17222  df-ress 17243  df-hom 17286  df-cco 17287  df-cat 17676  df-cid 17677  df-homf 17678  df-ssc 17819  df-resc 17820  df-subc 17821  df-func 17867

This theorem is referenced by:  funcres2  17907  funcres2c  17912  fthres2b  17941