Theorem cofuval2 17900

Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuval2.b	⊢ 𝐵 = (Base‘𝐶)
cofuval2.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
cofuval2.x	⊢ (𝜑 → 𝐻(𝐷 Func 𝐸)𝐾)

Assertion

Ref	Expression
cofuval2	⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦   𝑥,𝐾,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)

Proof of Theorem cofuval2

Step	Hyp	Ref	Expression
1		cofuval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		cofuval2.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 5120	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofuval2.x	. . . 4 ⊢ (𝜑 → 𝐻(𝐷 Func 𝐸)𝐾)
6		df-br 5120	. . . 4 ⊢ (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
8	1, 4, 7	cofuval 17895	. 2 ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)))⟩)
9		relfunc 17875	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
10		brrelex12 5706	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
11	9, 5, 10	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12		op1stg 8000	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14		relfunc 17875	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
15		brrelex12 5706	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
16	14, 2, 15	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17		op1stg 8000	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	13, 18	coeq12d 5844	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻 ∘ 𝐹))
20		op2ndg 8001	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
21	11, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22	21	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23	18	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
24	23	fveq1d 6878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
25	23	fveq1d 6878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
26	22, 24, 25	oveq123d 7426	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
27		op2ndg 8001	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	16, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
30	29	oveqd 7422	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
31	26, 30	coeq12d 5844	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))
32	31	mpoeq3dva 7484	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
33	19, 32	opeq12d 4857	. 2 ⊢ (𝜑 → ⟨((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)))⟩ = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
34	8, 33	eqtrd 2770	1 ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   class class class wbr 5119   ∘ ccom 5658  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228   Func cfunc 17867   ∘_func ccofu 17869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-map 8842  df-ixp 8912  df-func 17871  df-cofu 17873

This theorem is referenced by:  catcisolem  18123  funcrngcsetcALT  20601  fuco11a  49239