cofid1 - Mathbox for Zhi Wang

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Theorem cofid1 50324

Description: Express the object part of (𝐺 ∘_func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)

**Hypotheses**
Ref	Expression
cofid1a.i	⊢ 𝐼 = (id_func‘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 50285	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
3		cofid1.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4	3	func1st 50285	. . . 4 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
5	4	fveq1d 6888	. . 3 ⊢ (𝜑 → ((1^st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
6	2, 5	fveq12d 6893	. 2 ⊢ (𝜑 → ((1^st ‘⟨𝐾, 𝐿⟩)‘((1^st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
7		cofid1a.i	. . 3 ⊢ 𝐼 = (id_func‘𝐷)
8		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
9		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		df-br 5103	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	3, 10	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		df-br 5103	. . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
13	1, 12	sylib 221	. . 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 50322	. 2 ⊢ (𝜑 → ((1^st ‘⟨𝐾, 𝐿⟩)‘((1^st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = 𝑋)
16	6, 15	eqtr3d 2797	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 Basecbs 17393 Func cfunc 18035 id_funccidfu 18036 ∘_func ccofu 18037

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7424 df-oprab 7425 df-mpo 7426 df-1st 8002 df-2nd 8003 df-map 8853 df-ixp 8930 df-func 18039 df-idfu 18040 df-cofu 18041

This theorem is used by: (None)

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