Theorem isfunc 16524

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		fvexd 6203	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
4		simpl 473	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → 𝑑 = 𝐷)
5	4	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
6		isfunc.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐷)
7	5, 6	syl6eqr 2674	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
8		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
9		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
10	9	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
11		isfunc.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (Base‘𝐸)
12	10, 11	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
13	8, 12	feq23d 6040	. . . . . . . . . 10 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓:𝐵⟶𝐶))
14		fvex 6201	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) ∈ V
15	11, 14	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝐶 ∈ V
16		fvex 6201	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) ∈ V
17	6, 16	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
18	15, 17	elmap 7886	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝑓:𝐵⟶𝐶)
19	13, 18	syl6bbr 278	. . . . . . . . 9 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)))
20	8	sqxpeqd 5141	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21	20	ixpeq1d 7920	. . . . . . . . . . 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)))
22	9	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = (Hom ‘𝐸))
23		isfunc.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (Hom ‘𝐸)
24	22, 23	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = 𝐽)
25	24	oveqd 6667	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓‘(1st ‘𝑧))(Hom ‘𝑒)(𝑓‘(2nd ‘𝑧))) = ((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))))
26		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
27	26	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = (Hom ‘𝐷))
28		isfunc.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (Hom ‘𝐷)
29	27, 28	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = 𝐻)
30	29	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑑)‘𝑧) = (𝐻‘𝑧))
31	25, 30	oveq12d 6668	. . . . . . . . . . . 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)))
32	31	ixpeq2dv 7924	. . . . . . . . . . 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)))
33	21, 32	eqtrd 2656	. . . . . . . . . 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)))
34	33	eleq2d 2687	. . . . . . . . 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))))
35	26	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = (Id‘𝐷))
36		isfunc.1	. . . . . . . . . . . . . . 15 ⊢ 1 = (Id‘𝐷)
37	35, 36	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = 1 )
38	37	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑑)‘𝑥) = ( 1 ‘𝑥))
39	38	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((𝑥𝑔𝑥)‘( 1 ‘𝑥)))
40	9	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = (Id‘𝐸))
41		isfunc.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (Id‘𝐸)
42	40, 41	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = 𝐼)
43	42	fveq1d 6193	. . . . . . . . . . . 12 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑒)‘(𝑓‘𝑥)) = (𝐼‘(𝑓‘𝑥)))
44	39, 43	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓‘𝑥)) ↔ ((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥))))
45	29	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘𝑑)𝑦) = (𝑥𝐻𝑦))
46	29	oveqd 6667	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐻𝑧))
47	26	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = (comp‘𝐷))
48		isfunc.x	. . . . . . . . . . . . . . . . . . . 20 ⊢ · = (comp‘𝐷)
49	47, 48	syl6eqr 2674	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = · )
50	49	oveqd 6667	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
51	50	oveqd 6667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚) = (𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))
52	51	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)))
53	9	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = (comp‘𝐸))
54		isfunc.o	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑂 = (comp‘𝐸)
55	53, 54	syl6eqr 2674	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = 𝑂)
56	55	oveqd 6667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑒)(𝑓‘𝑧)) = (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧)))
57	56	oveqd 6667	. . . . . . . . . . . . . . . 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)))
58	52, 57	eqeq12d 2637	. . . . . . . . . . . . . . 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))))
59	46, 58	raleqbidv 3152	. . . . . . . . . . . . . 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))))
60	45, 59	raleqbidv 3152	. . . . . . . . . . . . 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))))
61	8, 60	raleqbidv 3152	. . . . . . . . . . . 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))))
62	8, 61	raleqbidv 3152	. . . . . . . . . . 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))))
63	44, 62	anbi12d 747	. . . . . . . . . 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)))))
64	8, 63	raleqbidv 3152	. . . . . . . . 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)))))
65	19, 34, 64	3anbi123d 1399	. . . . . . . 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))))))
66		df-3an 1039	. . . . . . . 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)))))
67	65, 66	syl6bb 276	. . . . . . 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))))))
68	3, 7, 67	sbcied2 3473	. . . . . 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))))))
69	68	opabbidv 4716	. . . . 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))))})
70		df-func 16518	. . . . 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))))})
71		ovex 6678	. . . . . . 7 ⊢ (𝐶 ↑𝑚 𝐵) ∈ V
72		snex 4908	. . . . . . . 8 ⊢ {𝑓} ∈ V
73		ovex 6678	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
74	73	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
75		ixpexg 7932	. . . . . . . . 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)
76	74, 75	ax-mp 5	. . . . . . . 8 ⊢ X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∈ V
77	72, 76	xpex 6962	. . . . . . 7 ⊢ ({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∈ V
78	71, 77	iunex 7147	. . . . . 6 ⊢ ∪ 𝑓 ∈ (𝐶 ↑𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) ∈ V
79		simpl 473	. . . . . . . . . 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))))
80	79	anim2i 593	. . . . . . . . 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)))))
81	80	2eximi 1763	. . . . . . . 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)))))
82		elopab 4983	. . . . . . . 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))))))
83		eliunxp 5259	. . . . . . . 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)))))
84	81, 82, 83	3imtr4i 281	. . . . . . 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))))
85	84	ssriv 3607	. . . . . 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)))
86	78, 85	ssexi 4803	. . . . 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
87	69, 70, 86	ovmpt2a 6791	. . . 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))))})
88	1, 2, 87	syl2anc 693	. . 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))))})
89	88	breqd 4664	. 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))
90		brabv 6699	. . . 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))
91		elex 3212	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) → 𝐹 ∈ V)
92		elex 3212	. . . . . 6 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) → 𝐺 ∈ V)
93	91, 92	anim12i 590	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐵) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧))) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
94	93	3adant3 1081	. . . 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))
95		simpl 473	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
96	95	eleq1d 2686	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝐹 ∈ (𝐶 ↑𝑚 𝐵)))
97		simpr 477	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
98	95	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘(1st ‘𝑧)) = (𝐹‘(1st ‘𝑧)))
99	95	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘(2nd ‘𝑧)) = (𝐹‘(2nd ‘𝑧)))
100	98, 99	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘(1st ‘𝑧))𝐽(𝑓‘(2nd ‘𝑧))) = ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))))
101	100	oveq1d 6665	. . . . . . . . 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)))
102	101	ixpeq2dv 7924	. . . . . . . 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)))
103	97, 102	eleq12d 2695	. . . . . . 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))))
104	97	oveqd 6667	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑥) = (𝑥𝐺𝑥))
105	104	fveq1d 6193	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = ((𝑥𝐺𝑥)‘( 1 ‘𝑥)))
106	95	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
107	106	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐼‘(𝑓‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
108	105, 107	eqeq12d 2637	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑥𝑔𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝑓‘𝑥)) ↔ ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥))))
109	97	oveqd 6667	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑧) = (𝑥𝐺𝑧))
110	109	fveq1d 6193	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)))
111	95	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑦) = (𝐹‘𝑦))
112	106, 111	opeq12d 4410	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
113	95	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑧) = (𝐹‘𝑧))
114	112, 113	oveq12d 6668	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩𝑂(𝑓‘𝑧)) = (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧)))
115	97	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
116	115	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑦𝑔𝑧)‘𝑛) = ((𝑦𝐺𝑧)‘𝑛))
117	97	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
118	117	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑥𝑔𝑦)‘𝑚) = ((𝑥𝐺𝑦)‘𝑚))
119	114, 116, 118	oveq123d 6671	. . . . . . . . . . . 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)))
120	110, 119	eqeq12d 2637	. . . . . . . . . . 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))))
121	120	2ralbidv 2989	. . . . . . . . . 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))))
122	121	2ralbidv 2989	. . . . . . . . 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))))
123	108, 122	anbi12d 747	. . . . . . . 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)))))
124	123	ralbidv 2986	. . . . . . 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)))))
125	96, 103, 124	3anbi123d 1399	. . . . . 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))))))
126	66, 125	syl5bbr 274	. . . . 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))))))
127		eqid 2622	. . . . 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))))}
128	126, 127	brabga 4989	. . . 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))))))
129	90, 94, 128	pm5.21nii 368	. . 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)))))
130	15, 17	elmap 7886	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐶)
131	130	3anbi1i 1253	. . 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)))))
132	129, 131	bitri 264	. 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)))))
133	89, 132	syl6bb 276	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435  {csn 4177  ⟨cop 4183  ∪ ciun 4520   class class class wbr 4653  {copab 4712   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Xcixp 7908  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326   Func cfunc 16514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ixp 7909  df-func 16518

This theorem is referenced by:  isfuncd  16525  funcf1  16526  funcixp  16527  funcid  16530  funcco  16531  idfucl  16541  cofucl  16548  funcres2b  16557  funcpropd  16560