Theorem cofuval2 18062

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 5104	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofuval2.x	. . . 4 ⊢ (𝜑 → 𝐻(𝐷 Func 𝐸)𝐾)
6		df-br 5104	. . . 4 ⊢ (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
8	1, 4, 7	cofuval 18057	. 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 18037	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
10		brrelex12 5703	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
11	9, 5, 10	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12		op1stg 8013	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
13	11, 12	syl 18	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14		relfunc 18037	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
15		brrelex12 5703	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
16	14, 2, 15	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17		op1stg 8013	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
18	16, 17	syl 18	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	13, 18	coeq12d 5842	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻 ∘ 𝐹))
20		op2ndg 8014	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
21	11, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22	21	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23	18	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
24	23	fveq1d 6887	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
25	23	fveq1d 6887	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
26	22, 24, 25	oveq123d 7441	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
27		op2ndg 8014	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	16, 27	syl 18	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
30	29	oveqd 7437	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
31	26, 30	coeq12d 5842	. . . 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 7497	. . 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 4841	. 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 2796	1 ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ∘ ccom 5655  Rel wrel 5656  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387   Func cfunc 18029   ∘_func ccofu 18031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-ixp 8926  df-func 18033  df-cofu 18035

This theorem is used by:  catcisolem  18285  funcrngcsetcALT  20893  cofidval  50226  cofuoppf  50257  fuco11a  50435