Theorem cofid1a 49143

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 6826	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘𝐼))
3	2	fveq1d 6824	. 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 17788	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
9		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
10	5	func1st2nd 49107	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
11	10	funcrcl2 49110	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 4, 11, 7	idfu1 17784	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
13	3, 8, 12	3eqtr3d 2774	1 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117   Func cfunc 17758  id_funccidfu 17759   ∘_func ccofu 17760

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-ixp 8822  df-func 17762  df-idfu 17763  df-cofu 17764

This theorem is referenced by:  cofid1  49145  cofidf2a  49148  uobeqw  49250