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Mirrors > Home > MPE Home > Th. List > evlf1 | Structured version Visualization version GIF version |
Description: Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
evlf1.e | ⊢ 𝐸 = (𝐶 evalF 𝐷) |
evlf1.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
evlf1.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
evlf1.b | ⊢ 𝐵 = (Base‘𝐶) |
evlf1.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
evlf1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
evlf1 | ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlf1.e | . . . 4 ⊢ 𝐸 = (𝐶 evalF 𝐷) | |
2 | evlf1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | evlf1.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
4 | evlf1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
5 | eqid 2728 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2728 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | eqid 2728 | . . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | evlfval 18209 | . . 3 ⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩) |
9 | ovex 7453 | . . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V | |
10 | 4 | fvexi 6911 | . . . . 5 ⊢ 𝐵 ∈ V |
11 | 9, 10 | mpoex 8084 | . . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V |
12 | 9, 10 | xpex 7755 | . . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V |
13 | 12, 12 | mpoex 8084 | . . . 4 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) ∈ V |
14 | 11, 13 | op1std 8003 | . . 3 ⊢ (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩ → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥))) |
15 | 8, 14 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥))) |
16 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹) | |
17 | 16 | fveq2d 6901 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹)) |
18 | simprr 772 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋) | |
19 | 17, 18 | fveq12d 6904 | . 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
20 | evlf1.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
21 | evlf1.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
22 | fvexd 6912 | . 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V) | |
23 | 15, 19, 20, 21, 22 | ovmpod 7573 | 1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⦋csb 3892 ⟨cop 4635 × cxp 5676 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1st c1st 7991 2nd c2nd 7992 Basecbs 17180 Hom chom 17244 compcco 17245 Catccat 17644 Func cfunc 17840 Nat cnat 17931 evalF cevlf 18201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-evlf 18205 |
This theorem is referenced by: evlfcllem 18213 evlfcl 18214 uncf1 18228 yonedalem3a 18266 yonedalem3b 18271 yonedainv 18273 yonffthlem 18274 |
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