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Mirrors > Home > MPE Home > Th. List > fuccoval | Structured version Visualization version GIF version |
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
fucco.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
fucco.n | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
fucco.a | ⊢ 𝐴 = (Base‘𝐶) |
fucco.o | ⊢ · = (comp‘𝐷) |
fucco.x | ⊢ ∙ = (comp‘𝑄) |
fucco.f | ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺)) |
fucco.g | ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻)) |
fuccoval.f | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fuccoval | ⊢ (𝜑 → ((𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucco.q | . . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
2 | fucco.n | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
3 | fucco.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
4 | fucco.o | . . 3 ⊢ · = (comp‘𝐷) | |
5 | fucco.x | . . 3 ⊢ ∙ = (comp‘𝑄) | |
6 | fucco.f | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺)) | |
7 | fucco.g | . . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fucco 17007 | . 2 ⊢ (𝜑 → (𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))) |
9 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
10 | 9 | fveq2d 6450 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
11 | 9 | fveq2d 6450 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
12 | 10, 11 | opeq12d 4644 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
13 | 9 | fveq2d 6450 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋)) |
14 | 12, 13 | oveq12d 6940 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥)) = (〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))) |
15 | 9 | fveq2d 6450 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
16 | 9 | fveq2d 6450 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋)) |
17 | 14, 15, 16 | oveq123d 6943 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
18 | fuccoval.f | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | ovexd 6956 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V) | |
20 | 8, 17, 18, 19 | fvmptd 6548 | 1 ⊢ (𝜑 → ((𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 〈cop 4403 ‘cfv 6135 (class class class)co 6922 1st c1st 7443 Basecbs 16255 compcco 16350 Nat cnat 16986 FuncCat cfuc 16987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-hom 16362 df-cco 16363 df-func 16903 df-nat 16988 df-fuc 16989 |
This theorem is referenced by: fuccocl 17009 fucass 17013 evlfcllem 17247 yonedalem3b 17305 |
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