Theorem cofu2a 49856

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 5111	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5111	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 221	. . 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 17944	. 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 6887	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (2nd ‘⟨𝑀, 𝑁⟩))
15	4, 7	cofucl 17946	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
16	13, 15	eqeltrrd 2864	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
17		df-br 5111	. . . . . . 7 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
18	16, 17	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
19	18	func2nd 49839	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	14, 19	eqtrd 2798	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑁)
21	20	oveqd 7429	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌) = (𝑋𝑁𝑌))
22	21	fveq1d 6885	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))𝑌)‘𝑅) = ((𝑋𝑁𝑌)‘𝑅))
23	5	func2nd 49839	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	2	func1st 49838	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
25	24	fveq1d 6885	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
26	24	fveq1d 6885	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
27	23, 25, 26	oveq123d 7433	. . 3 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
28	2	func2nd 49839	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	oveqd 7429	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
30	29	fveq1d 6885	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅))
31	27, 30	fveq12d 6890	. 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 2807	1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ⟨cop 4596   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322   Func cfunc 17912   ∘_func ccofu 17914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ixp 8897  df-cat 17725  df-cid 17726  df-func 17916  df-cofu 17918

This theorem is referenced by:  uptrlem1  49971  uptr2  49982