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Theorem evlf1 18181
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 2724 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2724 . . . 4 (comp‘𝐷) = (comp‘𝐷)
7 eqid 2724 . . . 4 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 18178 . . 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 7435 . . . . 5 (𝐶 Func 𝐷) ∈ V
104fvexi 6896 . . . . 5 𝐵 ∈ V
119, 10mpoex 8060 . . . 4 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
129, 10xpex 7734 . . . . 5 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1312, 12mpoex 8060 . . . 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 7979 . . 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 768 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
1716fveq2d 6886 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
18 simprr 770 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
1917, 18fveq12d 6889 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
20 evlf1.f . 2 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
21 evlf1.x . 2 (𝜑𝑋𝐵)
22 fvexd 6897 . 2 (𝜑 → ((1st𝐹)‘𝑋) ∈ V)
2315, 19, 20, 21, 22ovmpod 7553 1 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  csb 3886  cop 4627   × cxp 5665  cfv 6534  (class class class)co 7402  cmpo 7404  1st c1st 7967  2nd c2nd 7968  Basecbs 17149  Hom chom 17213  compcco 17214  Catccat 17613   Func cfunc 17809   Nat cnat 17900   evalF cevlf 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-evlf 18174
This theorem is referenced by:  evlfcllem  18182  evlfcl  18183  uncf1  18197  yonedalem3a  18235  yonedalem3b  18240  yonedainv  18242  yonffthlem  18243
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