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Theorem evlf2val 16682
 Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e 𝐸 = (𝐶 evalF 𝐷)
evlfval.c (𝜑𝐶 ∈ Cat)
evlfval.d (𝜑𝐷 ∈ Cat)
evlfval.b 𝐵 = (Base‘𝐶)
evlfval.h 𝐻 = (Hom ‘𝐶)
evlfval.o · = (comp‘𝐷)
evlfval.n 𝑁 = (𝐶 Nat 𝐷)
evlf2.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
evlf2.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
evlf2.x (𝜑𝑋𝐵)
evlf2.y (𝜑𝑌𝐵)
evlf2.l 𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)
evlf2val.a (𝜑𝐴 ∈ (𝐹𝑁𝐺))
evlf2val.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
evlf2val (𝜑 → (𝐴𝐿𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))

Proof of Theorem evlf2val
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . . 3 𝐸 = (𝐶 evalF 𝐷)
2 evlfval.c . . 3 (𝜑𝐶 ∈ Cat)
3 evlfval.d . . 3 (𝜑𝐷 ∈ Cat)
4 evlfval.b . . 3 𝐵 = (Base‘𝐶)
5 evlfval.h . . 3 𝐻 = (Hom ‘𝐶)
6 evlfval.o . . 3 · = (comp‘𝐷)
7 evlfval.n . . 3 𝑁 = (𝐶 Nat 𝐷)
8 evlf2.f . . 3 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 evlf2.g . . 3 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 evlf2.x . . 3 (𝜑𝑋𝐵)
11 evlf2.y . . 3 (𝜑𝑌𝐵)
12 evlf2.l . . 3 𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12evlf2 16681 . 2 (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
14 simprl 790 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑎 = 𝐴)
1514fveq1d 6105 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → (𝑎𝑌) = (𝐴𝑌))
16 simprr 792 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑔 = 𝐾)
1716fveq2d 6107 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → ((𝑋(2nd𝐹)𝑌)‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
1815, 17oveq12d 6567 . 2 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔)) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
19 evlf2val.a . 2 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
20 evlf2val.k . 2 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
21 ovex 6577 . . 3 ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)) ∈ V
2221a1i 11 . 2 (𝜑 → ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)) ∈ V)
2313, 18, 19, 20, 22ovmpt2d 6686 1 (𝜑 → (𝐴𝐿𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148   Func cfunc 16337   Nat cnat 16424   evalF cevlf 16672 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-evlf 16676 This theorem is referenced by:  evlfcllem  16684  evlfcl  16685  uncf2  16700  yonedalem3b  16742
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