cofu1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cofu1		Structured version Visualization version GIF version

Theorem cofu1 17791

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 17790	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
5	4	fveq1d 6824	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋))
6		eqid 2731	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		relfunc 17769	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
8		1st2ndbr 7974	. . . . 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 17773	. . 3 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
11		cofu2nd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12		fvco3 6921	. . 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 2766	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 2111 class class class wbr 5089 ∘ ccom 5618 Rel wrel 5619 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 Basecbs 17120 Func cfunc 17761 ∘_func ccofu 17763

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-ixp 8822 df-func 17765 df-cofu 17767

This theorem is referenced by: cofucl 17795 cofuass 17796 cofull 17843 cofth 17844 catciso 18018 1st2ndprf 18112 uncf1 18142 uncf2 18143 yonedalem21 18179 yonedalem22 18184 cofu1a 49205 cofid1a 49223 uptrar 49327 cofuswapf1 49405 prcofdiag1 49504 prcofdiag 49505 oppfdiag1 49525 oppfdiag 49527

W3C validator