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Theorem cofuval2 16594
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 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
cofuval2.x (𝜑𝐻(𝐷 Func 𝐸)𝐾)
Assertion
Ref Expression
cofuval2 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (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
StepHypRef Expression
1 cofuval2.b . . 3 𝐵 = (Base‘𝐶)
2 cofuval2.f . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 4686 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
42, 3sylib 208 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5 cofuval2.x . . . 4 (𝜑𝐻(𝐷 Func 𝐸)𝐾)
6 df-br 4686 . . . 4 (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
75, 6sylib 208 . . 3 (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
81, 4, 7cofuval 16589 . 2 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)))⟩)
9 relfunc 16569 . . . . . 6 Rel (𝐷 Func 𝐸)
10 brrelex12 5189 . . . . . 6 ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
119, 5, 10sylancr 696 . . . . 5 (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12 op1stg 7222 . . . . 5 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
1311, 12syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14 relfunc 16569 . . . . . 6 Rel (𝐶 Func 𝐷)
15 brrelex12 5189 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1614, 2, 15sylancr 696 . . . . 5 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17 op1stg 7222 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1816, 17syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1913, 18coeq12d 5319 . . 3 (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻𝐹))
20 op2ndg 7223 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
2111, 20syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22213ad2ant1 1102 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23183ad2ant1 1102 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2423fveq1d 6231 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹𝑥))
2523fveq1d 6231 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹𝑦))
2622, 24, 25oveq123d 6711 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
27 op2ndg 7223 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2816, 27syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29283ad2ant1 1102 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
3029oveqd 6707 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
3126, 30coeq12d 5319 . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))
3231mpt2eq3dva 6761 . . 3 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦))))
3319, 32opeq12d 4441 . 2 (𝜑 → ⟨((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)))⟩ = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
348, 33eqtrd 2685 1 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216   class class class wbr 4685  ccom 5147  Rel wrel 5148  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Basecbs 15904   Func cfunc 16561  func ccofu 16563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-func 16565  df-cofu 16567
This theorem is referenced by:  catcisolem  16803  funcrngcsetcALT  42324
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