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Theorem cofuval2 17778
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 5107 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
42, 3sylib 217 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5 cofuval2.x . . . 4 (𝜑𝐻(𝐷 Func 𝐸)𝐾)
6 df-br 5107 . . . 4 (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
75, 6sylib 217 . . 3 (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
81, 4, 7cofuval 17773 . 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 17753 . . . . . 6 Rel (𝐷 Func 𝐸)
10 brrelex12 5685 . . . . . 6 ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
119, 5, 10sylancr 588 . . . . 5 (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12 op1stg 7934 . . . . 5 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
1311, 12syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14 relfunc 17753 . . . . . 6 Rel (𝐶 Func 𝐷)
15 brrelex12 5685 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1614, 2, 15sylancr 588 . . . . 5 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17 op1stg 7934 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1816, 17syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1913, 18coeq12d 5821 . . 3 (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻𝐹))
20 op2ndg 7935 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
2111, 20syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22213ad2ant1 1134 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23183ad2ant1 1134 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2423fveq1d 6845 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹𝑥))
2523fveq1d 6845 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹𝑦))
2622, 24, 25oveq123d 7379 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
27 op2ndg 7935 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2816, 27syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29283ad2ant1 1134 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
3029oveqd 7375 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
3126, 30coeq12d 5821 . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))
3231mpoeq3dva 7435 . . 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 4839 . 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 2773 1 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3444  cop 4593   class class class wbr 5106  ccom 5638  Rel wrel 5639  cfv 6497  (class class class)co 7358  cmpo 7360  1st c1st 7920  2nd c2nd 7921  Basecbs 17088   Func cfunc 17745  func ccofu 17747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-ixp 8839  df-func 17749  df-cofu 17751
This theorem is referenced by:  catcisolem  18001  funcrngcsetcALT  46383
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