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Theorem cofu1 17936

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 17935	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
5	4	fveq1d 6883	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋))
6		eqid 2763	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		relfunc 17914	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
8		1st2ndbr 8035	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9	7, 2, 8	sylancr 598	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
10	1, 6, 9	funcf1 17918	. . 3 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
11		cofu2nd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12		fvco3 6981	. . 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 595	. 2 ⊢ (𝜑 → (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))
14	5, 13	eqtrd 2798	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 1570 ∈ wcel 2143 class class class wbr 5109 ∘ ccom 5665 Rel wrel 5666 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 Basecbs 17264 Func cfunc 17906 ∘_func ccofu 17908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-map 8822 df-ixp 8892 df-func 17910 df-cofu 17912

This theorem is referenced by: cofucl 17940 cofuass 17941 cofull 17988 cofth 17989 catciso 18163 1st2ndprf 18257 uncf1 18287 uncf2 18288 yonedalem21 18324 yonedalem22 18329 cofu1a 49892 cofid1a 49910 uptrar 50014 cofuswapf1 50092 prcofdiag1 50191 prcofdiag 50192 oppfdiag1 50212 oppfdiag 50214

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