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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 2724 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2724 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | eqid 2724 | . . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | evlfval 18178 | . . 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 7435 | . . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V | |
10 | 4 | fvexi 6896 | . . . . 5 ⊢ 𝐵 ∈ V |
11 | 9, 10 | mpoex 8060 | . . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V |
12 | 9, 10 | xpex 7734 | . . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V |
13 | 12, 12 | mpoex 8060 | . . . 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 7979 | . . 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 768 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹) | |
17 | 16 | fveq2d 6886 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹)) |
18 | simprr 770 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋) | |
19 | 17, 18 | fveq12d 6889 | . 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 6897 | . 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V) | |
23 | 15, 19, 20, 21, 22 | ovmpod 7553 | 1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⦋csb 3886 ⟨cop 4627 × cxp 5665 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 1st c1st 7967 2nd c2nd 7968 Basecbs 17149 Hom chom 17213 compcco 17214 Catccat 17613 Func cfunc 17809 Nat cnat 17900 evalF cevlf 18170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-evlf 18174 |
This theorem is referenced by: evlfcllem 18182 evlfcl 18183 uncf1 18197 yonedalem3a 18235 yonedalem3b 18240 yonedainv 18242 yonffthlem 18243 |
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