Theorem funcco 17818

Description: A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
funcco.b	⊢ 𝐵 = (Base‘𝐷)
funcco.h	⊢ 𝐻 = (Hom ‘𝐷)
funcco.o	⊢ · = (comp‘𝐷)
funcco.O	⊢ 𝑂 = (comp‘𝐸)
funcco.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
funcco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
funcco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
funcco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
funcco.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
funcco.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))

Assertion

Ref	Expression
funcco	⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀)))

Proof of Theorem funcco

Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcco.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
2		funcco.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2733	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		funcco.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐷)
5		eqid 2733	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2733	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2733	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		funcco.o	. . . . 5 ⊢ · = (comp‘𝐷)
9		funcco.O	. . . . 5 ⊢ 𝑂 = (comp‘𝐸)
10		df-br 5149	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 217	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 17810	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 496	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 497	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 17811	. . . 4 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(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))))))
17	1, 16	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(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)))))
18	17	simp3d 1145	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(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))))
19		funcco.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20		funcco.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
21	20	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
22		funcco.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23	22	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
24		funcco.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
25	24	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑀 ∈ (𝑋𝐻𝑌))
26		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
27		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
28	26, 27	oveq12d 7424	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
29	25, 28	eleqtrrd 2837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑀 ∈ (𝑥𝐻𝑦))
30		funcco.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))
31	30	ad4antr 731	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑁 ∈ (𝑌𝐻𝑍))
32		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑦 = 𝑌)
33		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑧 = 𝑍)
34	32, 33	oveq12d 7424	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
35	31, 34	eleqtrrd 2837	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑁 ∈ (𝑦𝐻𝑧))
36		simp-5r 785	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑥 = 𝑋)
37		simpllr 775	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑧 = 𝑍)
38	36, 37	oveq12d 7424	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑥𝐺𝑧) = (𝑋𝐺𝑍))
39		simp-4r 783	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑦 = 𝑌)
40	36, 39	opeq12d 4881	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
41	40, 37	oveq12d 7424	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
42		simpr 486	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
43		simplr 768	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀)
44	41, 42, 43	oveq123d 7427	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚) = (𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀))
45	38, 44	fveq12d 6896	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)))
46	36	fveq2d 6893	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑥) = (𝐹‘𝑋))
47	39	fveq2d 6893	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑦) = (𝐹‘𝑌))
48	46, 47	opeq12d 4881	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
49	37	fveq2d 6893	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑧) = (𝐹‘𝑍))
50	48, 49	oveq12d 7424	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧)) = (⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍)))
51	39, 37	oveq12d 7424	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑦𝐺𝑧) = (𝑌𝐺𝑍))
52	51, 42	fveq12d 6896	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑦𝐺𝑧)‘𝑛) = ((𝑌𝐺𝑍)‘𝑁))
53	36, 39	oveq12d 7424	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
54	53, 43	fveq12d 6896	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑥𝐺𝑦)‘𝑚) = ((𝑋𝐺𝑌)‘𝑀))
55	50, 52, 54	oveq123d 7427	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀)))
56	45, 55	eqeq12d 2749	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (((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))))
57	35, 56	rspcdv 3605	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → (∀𝑛 ∈ (𝑦𝐻𝑧)((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))))
58	29, 57	rspcimdv 3603	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((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))))
59	23, 58	rspcimdv 3603	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((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))))
60	21, 59	rspcimdv 3603	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 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)) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀))))
61	60	adantld 492	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(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))) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀))))
62	19, 61	rspcimdv 3603	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(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))) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀))))
63	18, 62	mpd 15	1 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ⟨cop 4634   class class class wbr 5148   × cxp 5674  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406  1^st c1st 7970  2^nd c2nd 7971   ↑_m cmap 8817  Xcixp 8888  Basecbs 17141  Hom chom 17205  compcco 17206  Catccat 17605  Idccid 17606   Func cfunc 17801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-map 8819  df-ixp 8889  df-func 17805

This theorem is referenced by:  funcsect  17819  funcoppc  17822  cofucl  17835  funcres  17843  fthsect  17873  fthmon  17875  catcisolem  18057  prfcl  18152  evlfcllem  18171  curf1cl  18178  curf2cl  18181  curfcl  18182  uncfcurf  18189  yonedalem4c  18227