Theorem cofid2a 49358

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 6838	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘𝐼))
3	2	oveqd 7375	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = (𝑋(2nd ‘𝐼)𝑌))
4	3	fveq1d 6836	. 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 17810	. 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 49321	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
15	14	funcrcl2 49324	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	13, 5, 15, 10, 8, 9, 11	idfu2 17802	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝑅) = 𝑅)
17	4, 12, 16	3eqtr3d 2779	1 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188   Func cfunc 17778  id_funccidfu 17779   ∘_func ccofu 17780

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-func 17782  df-idfu 17783  df-cofu 17784

This theorem is referenced by:  cofid2  49360