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Theorem evlf1 17529
 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 2759 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2759 . . . 4 (comp‘𝐷) = (comp‘𝐷)
7 eqid 2759 . . . 4 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 17526 . . 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 7184 . . . . 5 (𝐶 Func 𝐷) ∈ V
104fvexi 6673 . . . . 5 𝐵 ∈ V
119, 10mpoex 7783 . . . 4 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
129, 10xpex 7475 . . . . 5 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1312, 12mpoex 7783 . . . 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 7704 . . 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 6663 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → (1st𝑓) = (1st𝐹))
18 simprr 773 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → 𝑥 = 𝑋)
1917, 18fveq12d 6666 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑥 = 𝑋)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑋))
20 evlf1.f . 2 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
21 evlf1.x . 2 (𝜑𝑋𝐵)
22 fvexd 6674 . 2 (𝜑 → ((1st𝐹)‘𝑋) ∈ V)
2315, 19, 20, 21, 22ovmpod 7298 1 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ⦋csb 3806  ⟨cop 4529   × cxp 5523  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  1st c1st 7692  2nd c2nd 7693  Basecbs 16534  Hom chom 16627  compcco 16628  Catccat 16986   Func cfunc 17176   Nat cnat 17263   evalF cevlf 17518 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-evlf 17522 This theorem is referenced by:  evlfcllem  17530  evlfcl  17531  uncf1  17545  yonedalem3a  17583  yonedalem3b  17588  yonedainv  17590  yonffthlem  17591
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