Theorem cofid2 49402

Description: Express the morphism part of (𝐺 ∘_func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofid1a.i	⊢ 𝐼 = (idfunc‘𝐷)
cofid1a.b	⊢ 𝐵 = (Base‘𝐷)
cofid1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofid1.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
cofid1.k	⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
cofid1.o	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
cofid2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofid2.h	⊢ 𝐻 = (Hom ‘𝐷)
cofid2.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
cofid2	⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = 𝑅)

Proof of Theorem cofid2

Step	Hyp	Ref	Expression
1		cofid1.k	. . . . 5 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
2	1	func2nd 49365	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
3		cofid1.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4	3	func1st 49364	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
5	4	fveq1d 6837	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
6	4	fveq1d 6837	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
7	2, 5, 6	oveq123d 7381	. . 3 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
8	3	func2nd 49365	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
9	8	oveqd 7377	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
10	9	fveq1d 6837	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅))
11	7, 10	fveq12d 6842	. 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)))
12		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
13		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
14		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		df-br 5100	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
16	3, 15	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
17		df-br 5100	. . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
18	1, 17	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
19		cofid1.o	. . 3 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
20		cofid2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
21		cofid2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
22		cofid2.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
23	12, 13, 14, 16, 18, 19, 20, 21, 22	cofid2a 49400	. 2 ⊢ (𝜑 → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))‘((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌)‘𝑅)) = 𝑅)
24	11, 23	eqtr3d 2774	1 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4587   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17140  Hom chom 17192   Func cfunc 17782  id_funccidfu 17783   ∘_func ccofu 17784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-ixp 8840  df-func 17786  df-idfu 17787  df-cofu 17788

This theorem is referenced by: (None)