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Theorem evlf2val 17464
 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 17463 . 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 6651 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → (𝑎𝑌) = (𝐴𝑌))
16 simprr 772 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → 𝑔 = 𝐾)
1716fveq2d 6653 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑔 = 𝐾)) → ((𝑋(2nd𝐹)𝑌)‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝐾))
1815, 17oveq12d 7157 . 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 7174 . 2 (𝜑 → ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)) ∈ V)
2213, 18, 19, 20, 21ovmpod 7285 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 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⟨cop 4534  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Basecbs 16478  Hom chom 16571  compcco 16572  Catccat 16930   Func cfunc 17119   Nat cnat 17206   evalF cevlf 17454 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-evlf 17458 This theorem is referenced by:  evlfcllem  17466  evlfcl  17467  uncf2  17482  yonedalem3b  17524
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