Theorem cofu2a 49012

Description: Value of the morphism part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)

Hypotheses

Ref	Expression
cofu1a.b	⊢ 𝐵 = (Base‘𝐶)
cofu1a.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
cofu1a.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
cofu1a.m	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
cofu1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofu2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofu2a.h	⊢ 𝐻 = (Hom ‘𝐶)
cofu2a.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
cofu2a	⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))

Proof of Theorem cofu2a

Step	Hyp	Ref	Expression
1		cofu1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		cofu1a.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 5116	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5116	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
8		cofu1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		cofu2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		cofu2a.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
11		cofu2a.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
12	1, 4, 7, 8, 9, 10, 11	cofu2 17854	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌)‘𝑅) = ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))‘((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅)))
13		cofu1a.m	. . . . . 6 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
14	13	fveq2d 6869	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (2nd ‘⟨𝑀, 𝑁⟩))
15	4, 7	cofucl 17856	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
16	13, 15	eqeltrrd 2830	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
17		df-br 5116	. . . . . . 7 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
18	16, 17	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
19	18	func2nd 48995	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	14, 19	eqtrd 2765	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑁)
21	20	oveqd 7411	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌) = (𝑋𝑁𝑌))
22	21	fveq1d 6867	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌)‘𝑅) = ((𝑋𝑁𝑌)‘𝑅))
23	5	func2nd 48995	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	2	func1st 48994	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
25	24	fveq1d 6867	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
26	24	fveq1d 6867	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
27	23, 25, 26	oveq123d 7415	. . 3 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
28	2	func2nd 48995	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	oveqd 7411	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
30	29	fveq1d 6867	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅))
31	27, 30	fveq12d 6872	. 2 ⊢ (𝜑 → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))‘((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅)) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)))
32	12, 22, 31	3eqtr3rd 2774	1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4603   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237   Func cfunc 17822   ∘_func ccofu 17824

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 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828

This theorem is referenced by:  uptrlem1  49117