Theorem cofid1a 49237

Description: Express the object 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)

Assertion

Ref	Expression
cofid1a	⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Proof of Theorem cofid1a

Step	Hyp	Ref	Expression
1		cofid1a.o	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
2	1	fveq2d 6832	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘𝐼))
3	2	fveq1d 6830	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐼)‘𝑋))
4		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
5		cofid1a.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
6		cofid1a.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
7		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	4, 5, 6, 7	cofu1 17793	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
9		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
10	5	func1st2nd 49201	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
11	10	funcrcl2 49204	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 4, 11, 7	idfu1 17789	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
13	3, 8, 12	3eqtr3d 2776	1 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17122   Func cfunc 17763  id_funccidfu 17764   ∘_func ccofu 17765

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-func 17767  df-idfu 17768  df-cofu 17769

This theorem is referenced by:  cofid1  49239  cofidf2a  49242  uobeqw  49344