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Theorem evlf1 16683
 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 2610 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2610 . . . 4 (comp‘𝐷) = (comp‘𝐷)
7 eqid 2610 . . . 4 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 16680 . . 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 6577 . . . . 5 (𝐶 Func 𝐷) ∈ V
10 fvex 6113 . . . . . 6 (Base‘𝐶) ∈ V
114, 10eqeltri 2684 . . . . 5 𝐵 ∈ V
129, 11mpt2ex 7136 . . . 4 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
139, 11xpex 6860 . . . . 5 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1413, 13mpt2ex 7136 . . . 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
1512, 14op1std 7069 . . 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)))
168, 15syl 17 . 2 (𝜑 → (1st𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)))
17 simprl 790 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑓 = 𝐹)
1817fveq2d 6107 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
19 simprr 792 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
2018, 19fveq12d 6109 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
21 evlf1.f . 2 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
22 evlf1.x . 2 (𝜑𝑋𝐵)
23 fvex 6113 . . 3 ((1st𝐹)‘𝑋) ∈ V
2423a1i 11 . 2 (𝜑 → ((1st𝐹)‘𝑋) ∈ V)
2516, 20, 21, 22, 24ovmpt2d 6686 1 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148   Func cfunc 16337   Nat cnat 16424   evalF cevlf 16672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-evlf 16676 This theorem is referenced by:  evlfcllem  16684  evlfcl  16685  uncf1  16699  yonedalem3a  16737  yonedalem3b  16742  yonedainv  16744  yonffthlem  16745
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