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| Mirrors > Home > MPE Home > Th. List > uncfcl | Structured version Visualization version GIF version | ||
| Description: The uncurry operation takes a functor 𝐹:𝐶⟶(𝐷⟶𝐸) to a functor uncurryF (𝐹):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| uncfval.g | ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) | 
| uncfval.c | ⊢ (𝜑 → 𝐷 ∈ Cat) | 
| uncfval.d | ⊢ (𝜑 → 𝐸 ∈ Cat) | 
| uncfval.f | ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) | 
| Ref | Expression | 
|---|---|
| uncfcl | ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uncfval.g | . . 3 ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) | |
| 2 | uncfval.c | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
| 3 | uncfval.d | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) | |
| 4 | uncfval.f | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) | |
| 5 | 1, 2, 3, 4 | uncfval 18280 | . 2 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷)))) | 
| 6 | eqid 2736 | . . . 4 ⊢ ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷)) = ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷)) | |
| 7 | eqid 2736 | . . . 4 ⊢ ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷) | |
| 8 | eqid 2736 | . . . . . 6 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
| 9 | funcrcl 17909 | . . . . . . . 8 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat)) | |
| 10 | 4, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat)) | 
| 11 | 10 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 12 | eqid 2736 | . . . . . 6 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
| 13 | 8, 11, 2, 12 | 1stfcl 18243 | . . . . 5 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) | 
| 14 | 13, 4 | cofucl 17934 | . . . 4 ⊢ (𝜑 → (𝐺 ∘func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸))) | 
| 15 | eqid 2736 | . . . . 5 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷) | |
| 16 | 8, 11, 2, 15 | 2ndfcl 18244 | . . . 4 ⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) | 
| 17 | 6, 7, 14, 16 | prfcl 18249 | . . 3 ⊢ (𝜑 → ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷))) | 
| 18 | eqid 2736 | . . . 4 ⊢ (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸) | |
| 19 | eqid 2736 | . . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) | |
| 20 | 18, 19, 2, 3 | evlfcl 18268 | . . 3 ⊢ (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸)) | 
| 21 | 17, 20 | cofucl 17934 | . 2 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷))) ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | 
| 22 | 5, 21 | eqeltrd 2840 | 1 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 〈“cs3 14882 Catccat 17708 Func cfunc 17900 ∘func ccofu 17902 FuncCat cfuc 17991 ×c cxpc 18214 1stF c1stf 18215 2ndF c2ndf 18216 〈,〉F cprf 18217 evalF cevlf 18255 uncurryF cuncf 18257 | 
| 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 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 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-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 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-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 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-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-s2 14888 df-s3 14889 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-hom 17322 df-cco 17323 df-cat 17712 df-cid 17713 df-func 17904 df-cofu 17906 df-nat 17992 df-fuc 17993 df-xpc 18218 df-1stf 18219 df-2ndf 18220 df-prf 18221 df-evlf 18259 df-uncf 18261 | 
| This theorem is referenced by: curfuncf 18284 uncfcurf 18285 | 
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