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Theorem evlf1 18277
Description: Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlf1.e 𝐸 = (𝐶 evalF 𝐷)
evlf1.c (𝜑𝐶 ∈ Cat)
evlf1.d (𝜑𝐷 ∈ Cat)
evlf1.b 𝐵 = (Base‘𝐶)
evlf1.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
evlf1.x (𝜑𝑋𝐵)
Assertion
Ref Expression
evlf1 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))

Proof of Theorem evlf1
Dummy variables 𝑥 𝑦 𝑓 𝑎 𝑔 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlf1.e . . . 4 𝐸 = (𝐶 evalF 𝐷)
2 evlf1.c . . . 4 (𝜑𝐶 ∈ Cat)
3 evlf1.d . . . 4 (𝜑𝐷 ∈ Cat)
4 evlf1.b . . . 4 𝐵 = (Base‘𝐶)
5 eqid 2735 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2735 . . . 4 (comp‘𝐷) = (comp‘𝐷)
7 eqid 2735 . . . 4 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 18274 . . 3 (𝜑𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
9 ovex 7464 . . . . 5 (𝐶 Func 𝐷) ∈ V
104fvexi 6921 . . . . 5 𝐵 ∈ V
119, 10mpoex 8103 . . . 4 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
129, 10xpex 7772 . . . . 5 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1312, 12mpoex 8103 . . . 4 (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) ∈ V
1411, 13op1std 8023 . . 3 (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ → (1st𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)))
158, 14syl 17 . 2 (𝜑 → (1st𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)))
16 simprl 771 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
1716fveq2d 6911 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
18 simprr 773 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
1917, 18fveq12d 6914 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
20 evlf1.f . 2 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
21 evlf1.x . 2 (𝜑𝑋𝐵)
22 fvexd 6922 . 2 (𝜑 → ((1st𝐹)‘𝑋) ∈ V)
2315, 19, 20, 21, 22ovmpod 7585 1 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709   Func cfunc 17905   Nat cnat 17996   evalF cevlf 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-evlf 18270
This theorem is referenced by:  evlfcllem  18278  evlfcl  18279  uncf1  18293  yonedalem3a  18331  yonedalem3b  18336  yonedainv  18338  yonffthlem  18339
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