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Theorem cofu1st 18067

Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)

**Hypotheses**
Ref	Expression
cofuval.b	⊢ 𝐵 = (Base‘𝐶)
cofuval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

**Assertion**
Ref	Expression
cofu1st	⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))

Proof of Theorem cofu1st Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofuval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		cofuval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		cofuval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4	1, 2, 3	cofuval 18066	. . 3 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
5	4	fveq2d 6894	. 2 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = (1^st ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩))
6		fvex 6903	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
7		fvex 6903	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
8	6, 7	coex 7946	. . 3 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) ∈ V
9	1	fvexi 6904	. . . 4 ⊢ 𝐵 ∈ V
10	9, 9	mpoex 8097	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) ∈ V
11	8, 10	op1st 8013	. 2 ⊢ (1^st ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹))
12	5, 11	eqtrdi 2811	1 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 ∘ ccom 5655 ‘cfv 6543 (class class class)co 7424 ∈ cmpo 7426 1^st c1st 8003 2^nd c2nd 8004 Basecbs 17396 Func cfunc 18038 ∘_func ccofu 18040

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 5327 ax-pr 5391 ax-un 7755

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 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-map 8857 df-ixp 8934 df-func 18042 df-cofu 18044

This theorem is used by: cofu1 18068 cofucl 18072 cofuass 18073 catciso 18295 cofu1stfix 50348

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