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Theorem cofuval2 16370
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 4584 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
42, 3sylib 207 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5 cofuval2.x . . . 4 (𝜑𝐻(𝐷 Func 𝐸)𝐾)
6 df-br 4584 . . . 4 (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
75, 6sylib 207 . . 3 (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
81, 4, 7cofuval 16365 . 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 16345 . . . . . 6 Rel (𝐷 Func 𝐸)
10 brrelex12 5079 . . . . . 6 ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
119, 5, 10sylancr 694 . . . . 5 (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12 op1stg 7071 . . . . 5 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
1311, 12syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14 relfunc 16345 . . . . . 6 Rel (𝐶 Func 𝐷)
15 brrelex12 5079 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1614, 2, 15sylancr 694 . . . . 5 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17 op1stg 7071 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1816, 17syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1913, 18coeq12d 5208 . . 3 (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻𝐹))
20 op2ndg 7072 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
2111, 20syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22213ad2ant1 1075 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23183ad2ant1 1075 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2423fveq1d 6105 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹𝑥))
2523fveq1d 6105 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹𝑦))
2622, 24, 25oveq123d 6570 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
27 op2ndg 7072 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2816, 27syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29283ad2ant1 1075 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
3029oveqd 6566 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
3126, 30coeq12d 5208 . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))
3231mpt2eq3dva 6617 . . 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 4348 . 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 2644 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 1031   = wceq 1475  wcel 1977  Vcvv 3173  cop 4131   class class class wbr 4583  ccom 5042  Rel wrel 5043  cfv 5804  (class class class)co 6549  cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695   Func cfunc 16337  func ccofu 16339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-func 16341  df-cofu 16343
This theorem is referenced by:  catcisolem  16579  funcrngcsetcALT  41791
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