Theorem cofid2a 49400

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)
cofid1a.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofid1a.g	⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
cofid1a.o	⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
cofid2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofid2a.h	⊢ 𝐻 = (Hom ‘𝐷)
cofid2a.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
cofid2a	⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)

Proof of Theorem cofid2a

Step	Hyp	Ref	Expression
1		cofid1a.o	. . . . 5 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
2	1	fveq2d 6839	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘𝐼))
3	2	oveqd 7377	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = (𝑋(2nd ‘𝐼)𝑌))
4	3	fveq1d 6837	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((𝑋(2nd ‘𝐼)𝑌)‘𝑅))
5		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
6		cofid1a.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
7		cofid1a.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
8		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		cofid2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		cofid2a.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
11		cofid2a.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
12	5, 6, 7, 8, 9, 10, 11	cofu2 17814	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
13		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
14	6	func1st2nd 49363	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
15	14	funcrcl2 49366	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	13, 5, 15, 10, 8, 9, 11	idfu2 17806	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝑅) = 𝑅)
17	4, 12, 16	3eqtr3d 2780	1 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘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:  cofid2  49402