Theorem funcres2b 17167

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 5067	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2		funcrcl 17133	. . . . 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 5067	. . . . 5 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)))
7		funcrcl 17133	. . . . 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 2821	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	12, 13, 14	subcss1 17112	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
16	11, 15	fssd 6528	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐷))
17		eqid 2821	. . . . . . . . . 10 ⊢ (𝐷 ↾cat 𝑅) = (𝐷 ↾cat 𝑅)
18		subcrcl 17086	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
19	12, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	17, 14, 19, 13, 15	rescbas 17099	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘(𝐷 ↾cat 𝑅)))
21	20	feq3d 6501	. . . . . . . 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 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
27	26	frnd 6521	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
28	12	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
29	13	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	11	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶𝑆)
32		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
33	31, 32	ffvelrnd 6852	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ 𝑆)
34		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
35	31, 34	ffvelrnd 6852	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝑆)
36	28, 29, 30, 33, 35	subcss2 17113	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
37	27, 36	sstrd 3977	. . . . . . . . . . . . . 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 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 6359	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
41		df-f 6359	. . . . . . . . . . . 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 17100	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
44	43	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
45	44	oveqd 7173	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
46	45	feq3d 6501	. . . . . . . . . . 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 3191	. . . . . . . . 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 6670	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50		df-ov 7159	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
51	49, 50	syl6eqr 2874	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
52		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
54	52, 53	op1std 7699	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
55	54	fveq2d 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
56	52, 53	op2ndd 7700	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
57	56	fveq2d 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
58	55, 57	oveq12d 7174	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
59		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60		df-ov 7159	. . . . . . . . . . . . . . 15 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
61	59, 60	syl6eqr 2874	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
62	58, 61	oveq12d 7174	. . . . . . . . . . . . 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 2907	. . . . . . . . . . . 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 7189	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ∈ V
65		ovex 7189	. . . . . . . . . . . . 13 ⊢ (𝑥𝐻𝑦) ∈ V
66	64, 65	elmap 8435	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
67	63, 66	syl6bb 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 7174	. . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . 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 2907	. . . . . . . . . . . 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 7189	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ∈ V
72	71, 65	elmap 8435	. . . . . . . . . . . 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	syl6bb 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 348	. . . . . . . . . 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 5712	. . . . . . . . 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 3167	. . . . . . . 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 1438	. . . . . 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 8465	. . . . . 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 8465	. . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (Subcat‘𝐷))
84	13	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85		eqid 2821	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	11	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐹:𝐴⟶𝑆)
87	86	ffvelrnda 6851	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑆)
88	17, 83, 84, 85, 87	subcid 17117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → ((Id‘𝐷)‘(𝐹‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)))
89	88	eqeq2d 2832	. . . . . . 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 2821	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐷) = (comp‘𝐷)
91	17, 14, 19, 13, 15, 90	rescco 17102	. . . . . . . . . . . . 13 ⊢ (𝜑 → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
92	91	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
93	92	oveqd 7173	. . . . . . . . . . 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 7173	. . . . . . . . . 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 2832	. . . . . . . . 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 3199	. . . . . . . 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 3199	. . . . . . 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 3196	. . . . 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 1432	. . . 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 2821	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
104		eqid 2821	. . . . 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 17134	. . . 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 2821	. . . . 5 ⊢ (Base‘(𝐷 ↾cat 𝑅)) = (Base‘(𝐷 ↾cat 𝑅))
109		eqid 2821	. . . . 5 ⊢ (Hom ‘(𝐷 ↾cat 𝑅)) = (Hom ‘(𝐷 ↾cat 𝑅))
110		eqid 2821	. . . . 5 ⊢ (Id‘(𝐷 ↾cat 𝑅)) = (Id‘(𝐷 ↾cat 𝑅))
111		eqid 2821	. . . . 5 ⊢ (comp‘(𝐷 ↾cat 𝑅)) = (comp‘(𝐷 ↾cat 𝑅))
112	17, 12	subccat 17118	. . . . . 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 17134	. . . 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 383	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  ⟨cop 4573   class class class wbr 5066   × cxp 5553  ran crn 5556   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688   ↑_m cmap 8406  Xcixp 8461  Basecbs 16483  Hom chom 16576  compcco 16577  Catccat 16935  Idccid 16936   ↾_cat cresc 17078  Subcatcsubc 17079   Func cfunc 17124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-hom 16589  df-cco 16590  df-cat 16939  df-cid 16940  df-homf 16941  df-ssc 17080  df-resc 17081  df-subc 17082  df-func 17128

This theorem is referenced by:  funcres2  17168  funcres2c  17171  fthres2b  17200