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 17917

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 17916	. . 3 ⊢ (𝜑 → (1^st ‘(𝐺 ∘_func 𝐹)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
5	4	fveq1d 6869	. 2 ⊢ (𝜑 → ((1^st ‘(𝐺 ∘_func 𝐹))‘𝑋) = (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋))
6		eqid 2762	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
7		relfunc 17895	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
8		1st2ndbr 8023	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9	7, 2, 8	sylancr 596	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
10	1, 6, 9	funcf1 17899	. . 3 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
11		cofu2nd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12		fvco3 6967	. . 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 593	. 2 ⊢ (𝜑 → (((1^st ‘𝐺) ∘ (1^st ‘𝐹))‘𝑋) = ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)))
14	5, 13	eqtrd 2797	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 1560 ∈ wcel 2142 class class class wbr 5100 ∘ ccom 5651 Rel wrel 5652 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 Basecbs 17245 Func cfunc 17887 ∘_func ccofu 17889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-map 8810 df-ixp 8880 df-func 17891 df-cofu 17893

This theorem is referenced by: cofucl 17921 cofuass 17922 cofull 17969 cofth 17970 catciso 18144 1st2ndprf 18238 uncf1 18268 uncf2 18269 yonedalem21 18305 yonedalem22 18310 cofu1a 49715 cofid1a 49733 uptrar 49837 cofuswapf1 49915 prcofdiag1 50014 prcofdiag 50015 oppfdiag1 50035 oppfdiag 50037

W3C validator