Theorem cofid2a 49891

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 6885	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘𝐼))
3	2	oveqd 7427	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = (𝑋(2nd ‘𝐼)𝑌))
4	3	fveq1d 6883	. 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 17938	. 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 49854	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
15	14	funcrcl2 49857	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	13, 5, 15, 10, 8, 9, 11	idfu2 17930	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝑅) = 𝑅)
17	4, 12, 16	3eqtr3d 2806	1 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17264  Hom chom 17316   Func cfunc 17906  id_funccidfu 17907   ∘_func ccofu 17908

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-func 17910  df-idfu 17911  df-cofu 17912

This theorem is referenced by:  cofid2  49893