Theorem cofid1 49145

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)
cofid1.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
cofid1.k	⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
cofid1.o	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)

Assertion

Ref	Expression
cofid1	⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋)

Proof of Theorem cofid1

Step	Hyp	Ref	Expression
1		cofid1.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
2	1	func1st 49108	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
3		cofid1.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4	3	func1st 49108	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
5	4	fveq1d 6824	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
6	2, 5	fveq12d 6829	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
7		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
8		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
9		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		df-br 5092	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	3, 10	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		df-br 5092	. . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
13	1, 12	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
14		cofid1.o	. . 3 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
15	7, 8, 9, 11, 13, 14	cofid1a 49143	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = 𝑋)
16	6, 15	eqtr3d 2768	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4582   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  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: (None)