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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 2736 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 6 | eqid 2736 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
| 7 | eqid 2736 | . . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | evlfval 18263 | . . 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 7465 | . . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V | |
| 10 | 4 | fvexi 6919 | . . . . 5 ⊢ 𝐵 ∈ V | 
| 11 | 9, 10 | mpoex 8105 | . . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V | 
| 12 | 9, 10 | xpex 7774 | . . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V | 
| 13 | 12, 12 | mpoex 8105 | . . . 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 8025 | . . 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 6909 | . . 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 6912 | . 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 6920 | . 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V) | |
| 23 | 15, 19, 20, 21, 22 | ovmpod 7586 | 1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⦋csb 3898 〈cop 4631 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1st c1st 8013 2nd c2nd 8014 Basecbs 17248 Hom chom 17309 compcco 17310 Catccat 17708 Func cfunc 17900 Nat cnat 17990 evalF cevlf 18255 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-evlf 18259 | 
| This theorem is referenced by: evlfcllem 18267 evlfcl 18268 uncf1 18282 yonedalem3a 18320 yonedalem3b 18325 yonedainv 18327 yonffthlem 18328 | 
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