Theorem funcco 17840

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 2730	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		funcco.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐷)
5		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2730	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2730	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		funcco.o	. . . . 5 ⊢ · = (comp‘𝐷)
9		funcco.O	. . . . 5 ⊢ 𝑂 = (comp‘𝐸)
10		df-br 5111	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 218	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 17832	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 17833	. . . 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 232	. . 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 1144	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
22		funcco.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23	22	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
24		funcco.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
25	24	ad3antrrr 730	. . . . . . . 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 7408	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
29	25, 28	eleqtrrd 2832	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑀 ∈ (𝑥𝐻𝑦))
30		funcco.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))
31	30	ad4antr 732	. . . . . . . . 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 7408	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
35	31, 34	eleqtrrd 2832	. . . . . . . 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 7408	. . . . . . . . . 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 4848	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
41	40, 37	oveq12d 7408	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
42		simpr 484	. . . . . . . . . . 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 7411	. . . . . . . . . 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 6868	. . . . . . . . 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 6865	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑥) = (𝐹‘𝑋))
47	39	fveq2d 6865	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑦) = (𝐹‘𝑌))
48	46, 47	opeq12d 4848	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
49	37	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹‘𝑧) = (𝐹‘𝑍))
50	48, 49	oveq12d 7408	. . . . . . . . . 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 7408	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑦𝐺𝑧) = (𝑌𝐺𝑍))
52	51, 42	fveq12d 6868	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑦𝐺𝑧)‘𝑛) = ((𝑌𝐺𝑍)‘𝑁))
53	36, 39	oveq12d 7408	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
54	53, 43	fveq12d 6868	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑥𝐺𝑦)‘𝑚) = ((𝑋𝐺𝑌)‘𝑀))
55	50, 52, 54	oveq123d 7411	. . . . . . . . 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 2746	. . . . . . . 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 3583	. . . . . . 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 3581	. . . . . 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 3581	. . . . 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 3581	. . . 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 490	. . 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 3581	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟨cop 4598   class class class wbr 5110   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802  Xcixp 8873  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633   Func cfunc 17823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-ixp 8874  df-func 17827

This theorem is referenced by:  funcsect  17841  funcoppc  17844  cofucl  17857  funcres  17865  fthsect  17896  fthmon  17898  catcisolem  18079  prfcl  18171  evlfcllem  18189  curf1cl  18196  curf2cl  18199  curfcl  18200  uncfcurf  18207  yonedalem4c  18245  imaf1co  49148  fthcomf  49150  upciclem2  49160  uptrlem1  49203  fuco22natlem1  49335  fucocolem3  49348