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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 2733 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2733 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | eqid 2733 | . . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | evlfval 18111 | . . 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 7391 | . . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V | |
10 | 4 | fvexi 6857 | . . . . 5 ⊢ 𝐵 ∈ V |
11 | 9, 10 | mpoex 8013 | . . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V |
12 | 9, 10 | xpex 7688 | . . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V |
13 | 12, 12 | mpoex 8013 | . . . 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 7932 | . . 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 6847 | . . 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 6850 | . 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 6858 | . 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V) | |
23 | 15, 19, 20, 21, 22 | ovmpod 7508 | 1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⦋csb 3856 ⟨cop 4593 × cxp 5632 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 1st c1st 7920 2nd c2nd 7921 Basecbs 17088 Hom chom 17149 compcco 17150 Catccat 17549 Func cfunc 17745 Nat cnat 17833 evalF cevlf 18103 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-evlf 18107 |
This theorem is referenced by: evlfcllem 18115 evlfcl 18116 uncf1 18130 yonedalem3a 18168 yonedalem3b 18173 yonedainv 18175 yonffthlem 18176 |
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