Theorem cofuval2 17149

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 5031	. . . 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 5031	. . . 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 17144	. 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 17124	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
10		brrelex12 5568	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
11	9, 5, 10	sylancr 590	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12		op1stg 7683	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14		relfunc 17124	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
15		brrelex12 5568	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
16	14, 2, 15	sylancr 590	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17		op1stg 7683	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	13, 18	coeq12d 5699	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻 ∘ 𝐹))
20		op2ndg 7684	. . . . . . . 8 ⊢ ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
21	11, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22	21	3ad2ant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23	18	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
24	23	fveq1d 6647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
25	23	fveq1d 6647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
26	22, 24, 25	oveq123d 7156	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
27		op2ndg 7684	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	16, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	28	3ad2ant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
30	29	oveqd 7152	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
31	26, 30	coeq12d 5699	. . . 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 7210	. . 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 4773	. 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 2833	1 ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   class class class wbr 5030   ∘ ccom 5523  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1^st c1st 7669  2^nd c2nd 7670  Basecbs 16475   Func cfunc 17116   ∘_func ccofu 17118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-func 17120  df-cofu 17122

This theorem is referenced by:  catcisolem  17358  funcrngcsetcALT  44623