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Theorem cofu1st 17832

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 17831	. . 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 6895	. 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 6904	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
7		fvex 6904	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
8	6, 7	coex 7920	. . 3 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) ∈ V
9	1	fvexi 6905	. . . 4 ⊢ 𝐵 ∈ V
10	9, 9	mpoex 8065	. . 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 7982	. 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 2788	1 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟨cop 4634 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17143 Func cfunc 17803 ∘_func ccofu 17805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-ixp 8891 df-func 17807 df-cofu 17809

This theorem is referenced by: cofu1 17833 cofucl 17837 cofuass 17838 catciso 18060

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