Theorem cofu2a 49126

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 5092	. . . 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 5092	. . . 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 17790	. 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 6826	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (2nd ‘⟨𝑀, 𝑁⟩))
15	4, 7	cofucl 17792	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
16	13, 15	eqeltrrd 2832	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
17		df-br 5092	. . . . . . 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 49109	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	14, 19	eqtrd 2766	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑁)
21	20	oveqd 7363	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌) = (𝑋𝑁𝑌))
22	21	fveq1d 6824	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌)‘𝑅) = ((𝑋𝑁𝑌)‘𝑅))
23	5	func2nd 49109	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	2	func1st 49108	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
25	24	fveq1d 6824	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
26	24	fveq1d 6824	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
27	23, 25, 26	oveq123d 7367	. . 3 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
28	2	func2nd 49109	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	oveqd 7363	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
30	29	fveq1d 6824	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅))
31	27, 30	fveq12d 6829	. 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 2775	1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4582   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  Hom chom 17169   Func cfunc 17758   ∘_func ccofu 17760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-ixp 8822  df-cat 17571  df-cid 17572  df-func 17762  df-cofu 17764

This theorem is referenced by:  uptrlem1  49241  uptr2  49252