Theorem cofu2 17810

Description: Value of the morphism 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)
cofu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofu2.h	⊢ 𝐻 = (Hom ‘𝐶)
cofu2.y	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
cofu2	⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))

Proof of Theorem cofu2

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		cofu2nd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		cofu2nd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	cofu2nd 17809	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
7	6	fveq1d 6836	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅))
8		cofu2.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
9		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		relfunc 17786	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7986	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 2, 11	sylancr 587	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	1, 8, 9, 12, 4, 5	funcf2 17792	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌)))
14		cofu2.y	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
15		fvco3 6933	. . 3 ⊢ (((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌)) ∧ 𝑅 ∈ (𝑋𝐻𝑌)) → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
16	13, 14, 15	syl2anc 584	. 2 ⊢ (𝜑 → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
17	7, 16	eqtrd 2771	1 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   class class class wbr 5098   ∘ ccom 5628  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188   Func cfunc 17778   ∘_func ccofu 17780

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-func 17782  df-cofu 17784

This theorem is referenced by:  cofucl  17812  1st2ndprf  18129  uncf2  18160  yonedalem22  18201  cofu2a  49340  cofid2a  49358  cofuswapf2  49540  prcofdiag1  49638  prcofdiag  49639  oppfdiag  49661