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Theorem evlf2val 18159
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 18158 . 2 (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
14 simprl 770 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑎 = 𝐴)
1514fveq1d 6883 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → (𝑎𝑌) = (𝐴𝑌))
16 simprr 772 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑔 = 𝐾)
1716fveq2d 6885 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → ((𝑋(2nd𝐹)𝑌)‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
1815, 17oveq12d 7414 . 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 7431 . 2 (𝜑 → ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)) ∈ V)
2213, 18, 19, 20, 21ovmpod 7547 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 397   = wceq 1542  wcel 2107  Vcvv 3475  cop 4630  cfv 6535  (class class class)co 7396  1st c1st 7960  2nd c2nd 7961  Basecbs 17131  Hom chom 17195  compcco 17196  Catccat 17595   Func cfunc 17791   Nat cnat 17879   evalF cevlf 18149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-evlf 18153
This theorem is referenced by:  evlfcllem  18161  evlfcl  18162  uncf2  18177  yonedalem3b  18219
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