cofid1a - Mathbox for Zhi Wang

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Theorem cofid1a 50443

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)
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 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)) = 𝑋)

**Proof of Theorem cofid1a**
Step	Hyp	Ref	Expression
1		cofid1a.o	. . . 4 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐼)
2	1	fveq2d 6915	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = (1^st ‘𝐼))
3	2	fveq1d 6913	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = ((1^st ‘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 18091	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))
9		cofid1a.i	. . 3 ⊢ 𝐼 = (id_func‘𝐷)
10	5	func1st2nd 50405	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
11	10	funcrcl2 50408	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 4, 11, 7	idfu1 18087	. 2 ⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)
13	3, 8, 12	3eqtr3d 2806	1 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ‘cfv 6561 (class class class)co 7446 1^st c1st 8026 2^nd c2nd 8027 Basecbs 17419 Func cfunc 18061 id_funccidfu 18062 ∘_func ccofu 18063

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-map 8880 df-ixp 8957 df-func 18065 df-idfu 18066 df-cofu 18067

This theorem is used by: cofid1 50445 cofidf2a 50448 uobeqw 50550

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