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Theorem evlf2val 17067
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 17066 . 2 (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
14 simprl 754 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑎 = 𝐴)
1514fveq1d 6334 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → (𝑎𝑌) = (𝐴𝑌))
16 simprr 756 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑔 = 𝐾)
1716fveq2d 6336 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → ((𝑋(2nd𝐹)𝑌)‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
1815, 17oveq12d 6811 . 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 ovexd 6825 . 2 (𝜑 → ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)) ∈ V)
2213, 18, 19, 20, 21ovmpt2d 6935 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 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532   Func cfunc 16721   Nat cnat 16808   evalF cevlf 17057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-evlf 17061
This theorem is referenced by:  evlfcllem  17069  evlfcl  17070  uncf2  17085  yonedalem3b  17127
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