Theorem cofid1a 49029

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 6869	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘𝐼))
3	2	fveq1d 6867	. 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 17852	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
9		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
10	5	func1st2nd 48993	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
11	10	funcrcl2 48996	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 4, 11, 7	idfu1 17848	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
13	3, 8, 12	3eqtr3d 2773	1 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185   Func cfunc 17822  id_funccidfu 17823   ∘_func ccofu 17824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-func 17826  df-idfu 17827  df-cofu 17828

This theorem is referenced by:  cofid1  49031  cofidf2a  49034  uobeqw  49125