Theorem fucoisfunc2lem2 50694

Description: Lemma for fucoisfunc2 50695. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fucoisfunc.w	⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
fucoisfunc2lem1.o	⊢ 𝑂 = ( ∘func ↾ 𝑊)
fucoisfunc2lem1.b	⊢ 𝐵 = (Base‘𝐶)
fucoisfunc2lem1.r	⊢ 𝑅 = (Hom ‘𝐶)
fucoisfunc2lem1.s	⊢ 𝑆 = (Hom ‘𝐸)
fucoisfunc2lem1.v	⊢ ∙ = (comp‘𝐸)
fucoisfunc2.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
fucoisfunc2.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoisfunc2.h	⊢ 𝐻 = (Hom ‘𝑇)
fucoisfunc2.i	⊢ 𝐼 = (Id‘𝑇)
fucoisfunc2.x	⊢ · = (comp‘𝑇)
fucoisfunc2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucoisfunc2.d	⊢ (𝜑 → 𝐷 ∈ Cat)
fucoisfunc2.e	⊢ (𝜑 → 𝐸 ∈ Cat)
fucoisfunc2.k	⊢ 𝐾 = (Id‘𝐸)
fucoisfunc2lem2.j	⊢ 𝐽 = (Id‘𝑄)
fucoisfunc2lem2.y	⊢ ⚬ = (comp‘𝑄)

Assertion

Ref

Expression

fucoisfunc2lem2

⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))

Distinct variable groups:   𝐶,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐸,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑊,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑂,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,ℎ   𝑅,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,ℎ   𝑆,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,ℎ   ∙ ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,ℎ   𝐶,ℎ   𝐷,ℎ   ℎ,𝐸   ℎ,𝑊   ℎ,𝑂   𝑃,ℎ   𝑇,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝑄,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝐻,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝐼,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝐽,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝐾,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝜑,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   · ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   ⚬ ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧,𝑡,ℎ   𝑡,𝐵   𝑡,𝐶   𝑡,𝐷   𝑡,𝐸   𝑡,𝑊   𝑡,𝑂   𝑡,𝑃   𝑡,𝑅   𝑡,𝑆   𝑡, ∙

Proof of Theorem fucoisfunc2lem2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucoisfunc.w	. . . . 5 ⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
2		fucoisfunc2.t	. . . . 5 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
3		fucoisfunc2.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4		fucoisfunc2lem1.o	. . . . 5 ⊢ 𝑂 = ( ∘func ↾ 𝑊)
5		fucoisfunc2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝑇)
6		fucoisfunc2.i	. . . . 5 ⊢ 𝐼 = (Id‘𝑇)
7		fucoisfunc2lem2.j	. . . . 5 ⊢ 𝐽 = (Id‘𝑄)
8		fucoisfunc2.x	. . . . 5 ⊢ · = (comp‘𝑇)
9		fucoisfunc2lem2.y	. . . . 5 ⊢ ⚬ = (comp‘𝑄)
10		fucoisfunc2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		fucoisfunc2.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		fucoisfunc2.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	fucoisfunc 50692	. . . 4 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
14	2	fveq2i 6911	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
15	5, 14	eqtri 2786	. . . . . 6 ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
16	1, 15	fucofulem2 50638	. . . . 5 ⊢ (𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))))
17	16	anbi1i 636	. . . 4 ⊢ ((𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))) ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))))
18	13, 17	bitr4di 292	. . 3 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ (𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
19	2	xpcfucbas 50579	. . . . . 6 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
20	1, 19	eqtri 2786	. . . . 5 ⊢ 𝑊 = (Base‘𝑇)
21		fucoisfunc2lem1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
22		fucoisfunc2.k	. . . . 5 ⊢ 𝐾 = (Id‘𝐸)
23		fucoisfunc2lem1.v	. . . . 5 ⊢ ∙ = (comp‘𝐸)
24		eqid 2763	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
25	24, 11, 12	fuccat 18177	. . . . . . 7 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
26		eqid 2763	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
27	26, 10, 11	fuccat 18177	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
28	2, 25, 27	xpccat 18393	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ Cat)
29	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤))) → 𝑇 ∈ Cat)
30		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤))) → 𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)))
31	20, 3, 21, 5, 6, 7, 22, 8, 9, 23, 29, 30	fucidcobi 50552	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤))) → (∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))
32	31	pm5.32da 590	. . 3 ⊢ (𝜑 → ((𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))) ↔ (𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))
33	18, 32	bitrd 282	. 2 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ (𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))
34		fucoisfunc2lem1.r	. . . . 5 ⊢ 𝑅 = (Hom ‘𝐶)
35		fucoisfunc2lem1.s	. . . . 5 ⊢ 𝑆 = (Hom ‘𝐸)
36	1, 4, 21, 34, 35, 23	fucoisfunc2lem1 50693	. . . 4 ⊢ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
37	16, 36	bitri 278	. . 3 ⊢ (𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
38	37	anbi1i 636	. 2 ⊢ ((𝑃 ∈ X𝑤 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑤))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑤))) ↑m (𝐻‘𝑤)) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))) ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))
39	33, 38	bitrdi 290	1 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024   ↑_m cmap 8875  Xcixp 8953  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868   Func cfunc 18058   ∘_func ccofu 18060   Nat cnat 18148   FuncCat cfuc 18149   ×_c cxpc 18371

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-tp 4593  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-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  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-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-cofu 18064  df-nat 18150  df-fuc 18151  df-xpc 18375

This theorem is used by:  fucoisfunc2  50695