Theorem cofu2a 49220

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 5094	. . . 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 5094	. . . 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 17795	. 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 6832	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (2nd ‘⟨𝑀, 𝑁⟩))
15	4, 7	cofucl 17797	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
16	13, 15	eqeltrrd 2834	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
17		df-br 5094	. . . . . . 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 49203	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	14, 19	eqtrd 2768	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑁)
21	20	oveqd 7369	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌) = (𝑋𝑁𝑌))
22	21	fveq1d 6830	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌)‘𝑅) = ((𝑋𝑁𝑌)‘𝑅))
23	5	func2nd 49203	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	2	func1st 49202	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
25	24	fveq1d 6830	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
26	24	fveq1d 6830	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
27	23, 25, 26	oveq123d 7373	. . 3 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
28	2	func2nd 49203	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	oveqd 7369	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
30	29	fveq1d 6830	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅))
31	27, 30	fveq12d 6835	. 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 2777	1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4581   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17122  Hom chom 17174   Func cfunc 17763   ∘_func ccofu 17765

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-cat 17576  df-cid 17577  df-func 17767  df-cofu 17769

This theorem is referenced by:  uptrlem1  49335  uptr2  49346