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Theorem cofuval2 17157
 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 5053 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
42, 3sylib 221 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5 cofuval2.x . . . 4 (𝜑𝐻(𝐷 Func 𝐸)𝐾)
6 df-br 5053 . . . 4 (𝐻(𝐷 Func 𝐸)𝐾 ↔ ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
75, 6sylib 221 . . 3 (𝜑 → ⟨𝐻, 𝐾⟩ ∈ (𝐷 Func 𝐸))
81, 4, 7cofuval 17152 . 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 17132 . . . . . 6 Rel (𝐷 Func 𝐸)
10 brrelex12 5591 . . . . . 6 ((Rel (𝐷 Func 𝐸) ∧ 𝐻(𝐷 Func 𝐸)𝐾) → (𝐻 ∈ V ∧ 𝐾 ∈ V))
119, 5, 10sylancr 590 . . . . 5 (𝜑 → (𝐻 ∈ V ∧ 𝐾 ∈ V))
12 op1stg 7696 . . . . 5 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
1311, 12syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐻, 𝐾⟩) = 𝐻)
14 relfunc 17132 . . . . . 6 Rel (𝐶 Func 𝐷)
15 brrelex12 5591 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
1614, 2, 15sylancr 590 . . . . 5 (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
17 op1stg 7696 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1816, 17syl 17 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1913, 18coeq12d 5722 . . 3 (𝜑 → ((1st ‘⟨𝐻, 𝐾⟩) ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝐻𝐹))
20 op2ndg 7697 . . . . . . . 8 ((𝐻 ∈ V ∧ 𝐾 ∈ V) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
2111, 20syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
22213ad2ant1 1130 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐻, 𝐾⟩) = 𝐾)
23183ad2ant1 1130 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2423fveq1d 6663 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹𝑥))
2523fveq1d 6663 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹𝑦))
2622, 24, 25oveq123d 7170 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
27 op2ndg 7697 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2816, 27syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29283ad2ant1 1130 . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
3029oveqd 7166 . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
3126, 30coeq12d 5722 . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)(2nd ‘⟨𝐻, 𝐾⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) ∘ (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)) = (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))
3231mpoeq3dva 7224 . . 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 4797 . 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 2859 1 (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ⟨cop 4556   class class class wbr 5052   ∘ ccom 5546  Rel wrel 5547  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  1st c1st 7682  2nd c2nd 7683  Basecbs 16483   Func cfunc 17124   ∘func ccofu 17126 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-ixp 8458  df-func 17128  df-cofu 17130 This theorem is referenced by:  catcisolem  17366  funcrngcsetcALT  44549
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