Theorem funcres2b 18101

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 5107	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2		funcrcl 18067	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
3	1, 2	sylbi 220	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4	3	simpld 500	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat)
5	4	a1i 11	. 2 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 → 𝐶 ∈ Cat))
6		df-br 5107	. . . . 5 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)))
7		funcrcl 18067	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷 ↾cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
8	6, 7	sylbi 220	. . . 4 ⊢ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷 ↾cat 𝑅) ∈ Cat))
9	8	simpld 500	. . 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 2763	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
15	12, 13, 14	subcss1 18046	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐷))
16	11, 15	fssd 6749	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐷))
17		eqid 2763	. . . . . . . . . 10 ⊢ (𝐷 ↾cat 𝑅) = (𝐷 ↾cat 𝑅)
18		subcrcl 18020	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
19	12, 18	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	17, 14, 19, 13, 15	rescbas 18033	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘(𝐷 ↾cat 𝑅)))
21	20	feq3d 6716	. . . . . . . 8 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
22	11, 21	mpbid 235	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅)))
23	16, 22	2thd 268	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷 ↾cat 𝑅))))
24	23	adantr 486	. . . . 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 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
27	26	frnd 6740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦)))
28	12	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
29	13	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
31	11	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹:𝐴⟶𝑆)
32		simprl 783	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
33	31, 32	ffvelcdmd 7109	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑥) ∈ 𝑆)
34		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
35	31, 34	ffvelcdmd 7109	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝑆)
36	28, 29, 30, 33, 35	subcss2 18047	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
37	27, 36	sstrd 3923	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
38	37, 27	2thd 268	. . . . . . . . . . . . 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 642	. . . . . . . . . . . 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 6562	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
41		df-f 6562	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹‘𝑥)𝑅(𝐹‘𝑦))))
42	39, 40, 41	3bitr4g 317	. . . . . . . . . . 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 18034	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
44	43	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑅 = (Hom ‘(𝐷 ↾cat 𝑅)))
45	44	oveqd 7460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)))
46	45	feq3d 6716	. . . . . . . . . . 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 282	. . . . . . . . . 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 3209	. . . . . . . . 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 6908	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50		df-ov 7446	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
51	49, 50	eqtr4di 2816	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
52		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
54	52, 53	op1std 8035	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
55	54	fveq2d 6912	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
56	52, 53	op2ndd 8036	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
57	56	fveq2d 6912	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
58	55, 57	oveq12d 7461	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))(Hom ‘𝐷)(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
59		fveq2 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60		df-ov 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
61	59, 60	eqtr4di 2816	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
62	58, 61	oveq12d 7461	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 7476	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ∈ V
65		ovex 7476	. . . . . . . . . . . . 13 ⊢ (𝑥𝐻𝑦) ∈ V
66	64, 65	elmap 8927	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
67	63, 66	bitrdi 290	. . . . . . . . . . 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 7461	. . . . . . . . . . . . . 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 7461	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 7476	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥)(Hom ‘(𝐷 ↾cat 𝑅))(𝐹‘𝑦)) ∈ V
72	71, 65	elmap 8927	. . . . . . . . . . . 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 290	. . . . . . . . . . 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 5824	. . . . . . . . 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 237	. . . . . . . 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 3121	. . . . . . . 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 18	. . . . . . 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 1471	. . . . . 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 8957	. . . . . 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 8957	. . . . . 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 317	. . . . 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 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (Subcat‘𝐷))
84	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85		eqid 2763	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
86	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐹:𝐴⟶𝑆)
87	86	ffvelcdmda 7108	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑆)
88	17, 83, 84, 85, 87	subcid 18051	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → ((Id‘𝐷)‘(𝐹‘𝑥)) = ((Id‘(𝐷 ↾cat 𝑅))‘(𝐹‘𝑥)))
89	88	eqeq2d 2774	. . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐷) = (comp‘𝐷)
91	17, 14, 19, 13, 15, 90	rescco 18036	. . . . . . . . . . . . 13 ⊢ (𝜑 → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
92	91	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ 𝐴) → (comp‘𝐷) = (comp‘(𝐷 ↾cat 𝑅)))
93	92	oveqd 7460	. . . . . . . . . . 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 7460	. . . . . . . . . 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 2774	. . . . . . . . 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 3230	. . . . . . . 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 3230	. . . . . . 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 644	. . . . . 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 3187	. . . . 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 1465	. . . 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 2763	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
104		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
105		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
106	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
107	101, 14, 102, 30, 103, 85, 104, 90, 105, 106	isfunc 18068	. . . 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 2763	. . . . 5 ⊢ (Base‘(𝐷 ↾cat 𝑅)) = (Base‘(𝐷 ↾cat 𝑅))
109		eqid 2763	. . . . 5 ⊢ (Hom ‘(𝐷 ↾cat 𝑅)) = (Hom ‘(𝐷 ↾cat 𝑅))
110		eqid 2763	. . . . 5 ⊢ (Id‘(𝐷 ↾cat 𝑅)) = (Id‘(𝐷 ↾cat 𝑅))
111		eqid 2763	. . . . 5 ⊢ (comp‘(𝐷 ↾cat 𝑅)) = (comp‘(𝐷 ↾cat 𝑅))
112	17, 12	subccat 18052	. . . . . 6 ⊢ (𝜑 → (𝐷 ↾cat 𝑅) ∈ Cat)
113	112	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐷 ↾cat 𝑅) ∈ Cat)
114	101, 108, 102, 109, 103, 110, 104, 111, 105, 113	isfunc 18068	. . . 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 314	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))
116	115	ex 418	. 2 ⊢ (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺)))
117	5, 10, 116	pm5.21ndd 382	1 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106   × cxp 5653  ran crn 5656   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024   ↑_m cmap 8875  Xcixp 8953  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868   ↾_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-ssc 18014  df-resc 18015  df-subc 18016  df-func 18062

This theorem is used by:  funcres2  18102  funcres2c  18107  fthres2b  18136