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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 2735 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2735 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | eqid 2735 | . . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | evlfval 18274 | . . 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 7464 | . . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V | |
10 | 4 | fvexi 6921 | . . . . 5 ⊢ 𝐵 ∈ V |
11 | 9, 10 | mpoex 8103 | . . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V |
12 | 9, 10 | xpex 7772 | . . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V |
13 | 12, 12 | mpoex 8103 | . . . 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 8023 | . . 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 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹) | |
17 | 16 | fveq2d 6911 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹)) |
18 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋) | |
19 | 17, 18 | fveq12d 6914 | . 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 6922 | . 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V) | |
23 | 15, 19, 20, 21, 22 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘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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