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Theorem cofuval2 17833
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 5148 . . . 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 5148 . . . 4 (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
75, 6sylib 217 . . 3 (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
81, 4, 7cofuval 17828 . 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 17808 . . . . . 6 Rel (𝐷 Func 𝐸)
10 brrelex12 5726 . . . . . 6 ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
119, 5, 10sylancr 587 . . . . 5 (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12 op1stg 7983 . . . . 5 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
1311, 12syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14 relfunc 17808 . . . . . 6 Rel (𝐶 Func 𝐷)
15 brrelex12 5726 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1614, 2, 15sylancr 587 . . . . 5 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17 op1stg 7983 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1816, 17syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1913, 18coeq12d 5862 . . 3 (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻𝐹))
20 op2ndg 7984 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
2111, 20syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22213ad2ant1 1133 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23183ad2ant1 1133 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2423fveq1d 6890 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹𝑥))
2523fveq1d 6890 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹𝑦))
2622, 24, 25oveq123d 7426 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
27 op2ndg 7984 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2816, 27syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29283ad2ant1 1133 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
3029oveqd 7422 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
3126, 30coeq12d 5862 . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))
3231mpoeq3dva 7482 . . 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 4880 . 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 2772 1 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  cop 4633   class class class wbr 5147  ccom 5679  Rel wrel 5680  cfv 6540  (class class class)co 7405  cmpo 7407  1st c1st 7969  2nd c2nd 7970  Basecbs 17140   Func cfunc 17800  func ccofu 17802
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-func 17804  df-cofu 17806
This theorem is referenced by:  catcisolem  18056  funcrngcsetcALT  46850
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