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Theorem cofu1 17806

Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)

**Hypotheses**
Ref	Expression
cofuval.b	⊢ 𝐵 = (Base‘𝐶)
cofuval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
cofu1	⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))

**Proof of Theorem cofu1**
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	cofu1st 17805	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
5	4	fveq1d 6834	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋))
6		eqid 2734	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		relfunc 17784	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
8		1st2ndbr 7984	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9	7, 2, 8	sylancr 587	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
10	1, 6, 9	funcf1 17788	. . 3 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
11		cofu2nd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12		fvco3 6931	. . 3 ⊢ (((1^st ‘𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋 ∈ 𝐵) → (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))
13	10, 11, 12	syl2anc 584	. 2 ⊢ (𝜑 → (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))
14	5, 13	eqtrd 2769	1 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 class class class wbr 5096 ∘ ccom 5626 Rel wrel 5627 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17134 Func cfunc 17776 ∘_func ccofu 17778

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-map 8763 df-ixp 8834 df-func 17780 df-cofu 17782

This theorem is referenced by: cofucl 17810 cofuass 17811 cofull 17858 cofth 17859 catciso 18033 1st2ndprf 18127 uncf1 18157 uncf2 18158 yonedalem21 18194 yonedalem22 18199 cofu1a 49281 cofid1a 49299 uptrar 49403 cofuswapf1 49481 prcofdiag1 49580 prcofdiag 49581 oppfdiag1 49601 oppfdiag 49603

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