Theorem cofid2a 49695

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 6866	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘𝐼))
3	2	oveqd 7408	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = (𝑋(2nd ‘𝐼)𝑌))
4	3	fveq1d 6864	. 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 17910	. 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 49658	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
15	14	funcrcl2 49661	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	13, 5, 15, 10, 8, 9, 11	idfu2 17902	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝑅) = 𝑅)
17	4, 12, 16	3eqtr3d 2804	1 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6516  (class class class)co 7391  1^st c1st 7963  2^nd c2nd 7964  Basecbs 17236  Hom chom 17288   Func cfunc 17878  id_funccidfu 17879   ∘_func ccofu 17880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-ixp 8874  df-func 17882  df-idfu 17883  df-cofu 17884

This theorem is referenced by:  cofid2  49697