Theorem isfunc 16347

Description: Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
isfunc.b	⊢ 𝐵 = (Base‘𝐷)
isfunc.c	⊢ 𝐶 = (Base‘𝐸)
isfunc.h	⊢ 𝐻 = (Hom ‘𝐷)
isfunc.j	⊢ 𝐽 = (Hom ‘𝐸)
isfunc.1	⊢ 1 = (Id‘𝐷)
isfunc.i	⊢ 𝐼 = (Id‘𝐸)
isfunc.x	⊢ · = (comp‘𝐷)
isfunc.o	⊢ 𝑂 = (comp‘𝐸)
isfunc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
isfunc.e	⊢ (𝜑 → 𝐸 ∈ Cat)

Assertion

Ref	Expression
isfunc	⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))

Distinct variable groups:   𝑚,𝑛,𝑥,𝑦,𝑧,𝐵   𝐷,𝑚,𝑛,𝑥,𝑦,𝑧   𝑚,𝐸,𝑛,𝑥,𝑦,𝑧   𝑚,𝐻,𝑛,𝑥,𝑦,𝑧   𝑚,𝐹,𝑛,𝑥,𝑦,𝑧   𝑚,𝐺,𝑛,𝑥,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝜑,𝑚,𝑛,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑚,𝑛)   · (𝑥,𝑦,𝑧,𝑚,𝑛)   1 (𝑥,𝑦,𝑧,𝑚,𝑛)   𝐼(𝑥,𝑦,𝑧,𝑚,𝑛)   𝐽(𝑚,𝑛)   𝑂(𝑥,𝑦,𝑧,𝑚,𝑛)

Proof of Theorem isfunc

Dummy variables 𝑏 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfunc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
2		isfunc.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
3		fvex 6113	. . . . . . . 8 ⊢ (Base‘𝑑) ∈ V
4	3	a1i 11	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
5		simpl 472	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → 𝑑 = 𝐷)
6	5	fveq2d 6107	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
7		isfunc.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐷)
8	6, 7	syl6eqr 2662	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
9		simpr 476	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
10		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
11	10	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
12		isfunc.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (Base‘𝐸)
13	11, 12	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
14	9, 13	feq23d 5953	. . . . . . . . . 10 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓:𝐵⟶𝐶))
15		fvex 6113	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) ∈ V
16	12, 15	eqeltri 2684	. . . . . . . . . . 11 ⊢ 𝐶 ∈ V
17		fvex 6113	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) ∈ V
18	7, 17	eqeltri 2684	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
19	16, 18	elmap 7772	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝑓:𝐵⟶𝐶)
20	14, 19	syl6bbr 277	. . . . . . . . 9 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)))
21	9	sqxpeqd 5065	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
22	21	ixpeq1d 7806	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)))
23	10	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = (Hom ‘𝐸))
24		isfunc.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (Hom ‘𝐸)
25	23, 24	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = 𝐽)
26	25	oveqd 6566	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) = ((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))))
27		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
28	27	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = (Hom ‘𝐷))
29		isfunc.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (Hom ‘𝐷)
30	28, 29	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = 𝐻)
31	30	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑑)‘𝑧) = (𝐻‘𝑧))
32	26, 31	oveq12d 6567	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = (((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
33	32	ixpeq2dv 7810	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
34	22, 33	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
35	34	eleq2d 2673	. . . . . . . . 9 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ↔ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))))
36	27	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = (Id‘𝐷))
37		isfunc.1	. . . . . . . . . . . . . . 15 ⊢ 1 = (Id‘𝐷)
38	36, 37	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = 1 )
39	38	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑑)‘𝑥) = ( 1 ‘𝑥))
40	39	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((𝑥𝑔𝑥)‘( 1 ‘𝑥)))
41	10	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = (Id‘𝐸))
42		isfunc.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (Id‘𝐸)
43	41, 42	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = 𝐼)
44	43	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑒)‘(𝑓‘𝑥)) = (𝐼‘(𝑓‘𝑥)))
45	40, 44	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ↔ ((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥))))
46	30	oveqd 6566	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘𝑑)𝑦) = (𝑥𝐻𝑦))
47	30	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐻𝑧))
48	27	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = (comp‘𝐷))
49		isfunc.x	. . . . . . . . . . . . . . . . . . . 20 ⊢ · = (comp‘𝐷)
50	48, 49	syl6eqr 2662	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = · )
51	50	oveqd 6566	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
52	51	oveqd 6566	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚) = (𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))
53	52	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)))
54	10	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = (comp‘𝐸))
55		isfunc.o	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑂 = (comp‘𝐸)
56	54, 55	syl6eqr 2662	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = 𝑂)
57	56	oveqd 6566	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧)) = (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧)))
58	57	oveqd 6566	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))
59	53, 58	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))
60	47, 59	raleqbidv 3129	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))
61	46, 60	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))
62	9, 61	raleqbidv 3129	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))
63	9, 62	raleqbidv 3129	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))
64	45, 63	anbi12d 743	. . . . . . . . . 10 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))))
65	9, 64	raleqbidv 3129	. . . . . . . . 9 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))))
66	20, 35, 65	3anbi123d 1391	. . . . . . . 8 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))))
67		df-3an 1033	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))))
68	66, 67	syl6bb 275	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))))
69	4, 8, 68	sbcied2 3440	. . . . . 6 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ([(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))))
70	69	opabbidv 4648	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
71		df-func 16341	. . . . 5 ⊢ Func = (𝑑 ∈ Cat, 𝑒 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
72		ovex 6577	. . . . . . 7 ⊢ (𝐶 ↑𝑚 𝐵) ∈ V
73		snex 4835	. . . . . . . 8 ⊢ {𝑓} ∈ V
74		ovex 6577	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
75	74	rgenw 2908	. . . . . . . . 9 ⊢ ∀𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
76		ixpexg 7818	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V)
77	75, 76	ax-mp 5	. . . . . . . 8 ⊢ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
78	73, 77	xpex 6860	. . . . . . 7 ⊢ ({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∈ V
79	72, 78	iunex 7039	. . . . . 6 ⊢ ∪ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∈ V
80		simpl 472	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) → (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))))
81	80	anim2i 591	. . . . . . . . 9 ⊢ ((𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))) → (𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))))
82	81	2eximi 1753	. . . . . . . 8 ⊢ (∃𝑓∃𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))) → ∃𝑓∃𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))))
83		elopab 4908	. . . . . . . 8 ⊢ (𝑑 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} ↔ ∃𝑓∃𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))))
84		eliunxp 5181	. . . . . . . 8 ⊢ (𝑑 ∈ ∪ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ↔ ∃𝑓∃𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))))
85	82, 83, 84	3imtr4i 280	. . . . . . 7 ⊢ (𝑑 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} → 𝑑 ∈ ∪ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))))
86	85	ssriv 3572	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} ⊆ ∪ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
87	79, 86	ssexi 4731	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} ∈ V
88	70, 71, 87	ovmpt2a 6689	. . . 4 ⊢ ((𝐷 ∈ Cat ∧ 𝐸 ∈ Cat) → (𝐷 Func 𝐸) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
89	1, 2, 88	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
90	89	breqd 4594	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺))
91		brabv 6597	. . . 4 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
92		elex 3185	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) → 𝐹 ∈ V)
93		elex 3185	. . . . . 6 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) → 𝐺 ∈ V)
94	92, 93	anim12i 588	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
95	94	3adant3 1074	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
96		simpl 472	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
97	96	eleq1d 2672	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝐹 ∈ (𝐶 ↑𝑚 𝐵)))
98		simpr 476	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
99	96	fveq1d 6105	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘(1st ‘𝑧)) = (𝐹‘(1st ‘𝑧)))
100	96	fveq1d 6105	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘(2nd ‘𝑧)) = (𝐹‘(2nd ‘𝑧)))
101	99, 100	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) = ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))))
102	101	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) = (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
103	102	ixpeq2dv 7810	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
104	98, 103	eleq12d 2682	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ↔ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))))
105	98	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑥) = (𝑥𝐺𝑥))
106	105	fveq1d 6105	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = ((𝑥𝐺𝑥)‘( 1 ‘𝑥)))
107	96	fveq1d 6105	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
108	107	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐼‘(𝑓‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
109	106, 108	eqeq12d 2625	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ↔ ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥))))
110	98	oveqd 6566	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑧) = (𝑥𝐺𝑧))
111	110	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)))
112	96	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑦) = (𝐹‘𝑦))
113	107, 112	opeq12d 4348	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
114	96	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑧) = (𝐹‘𝑧))
115	113, 114	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧)) = (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧)))
116	98	oveqd 6566	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
117	116	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑦𝑔𝑧)‘𝑛) = ((𝑦𝐺𝑧)‘𝑛))
118	98	oveqd 6566	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
119	118	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑦)‘𝑚) = ((𝑥𝐺𝑦)‘𝑚))
120	115, 117, 119	oveq123d 6570	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))
121	111, 120	eqeq12d 2625	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
122	121	2ralbidv 2972	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
123	122	2ralbidv 2972	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
124	109, 123	anbi12d 743	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
125	124	ralbidv 2969	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
126	97, 104, 125	3anbi123d 1391	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
127	67, 126	syl5bbr 273	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
128		eqid 2610	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}
129	127, 128	brabga 4914	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
130	91, 95, 129	pm5.21nii 367	. . 3 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
131	16, 18	elmap 7772	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐶)
132	131	3anbi1i 1246	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))) ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
133	130, 132	bitri 263	. 2 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝑔 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
134	90, 133	syl6bb 275	1 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583  {copab 4642   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Xcixp 7794  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149   Func cfunc 16337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ixp 7795  df-func 16341

This theorem is referenced by:  isfuncd  16348  funcf1  16349  funcixp  16350  funcid  16353  funcco  16354  idfucl  16364  cofucl  16371  funcres2b  16380  funcpropd  16383