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Mirrors > Home > MPE Home > Th. List > fucbas | Structured version Visualization version GIF version |
Description: The objects of the functor category are functors from 𝐶 to 𝐷. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
fucbas.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
Ref | Expression |
---|---|
fucbas | ⊢ (𝐶 Func 𝐷) = (Base‘𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucbas.q | . . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
2 | eqid 2726 | . . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷) | |
3 | eqid 2726 | . . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷) | |
4 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | eqid 2726 | . . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
6 | simpl 481 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat) | |
7 | simpr 483 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat) | |
8 | eqid 2726 | . . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fuccofval 17976 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (comp‘𝑄) = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝐶 Nat 𝐷)ℎ), 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝐷)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | fucval 17975 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), (𝐶 Nat 𝐷)〉, 〈(comp‘ndx), (comp‘𝑄)〉}) |
11 | catstr 17974 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), (𝐶 Nat 𝐷)〉, 〈(comp‘ndx), (comp‘𝑄)〉} Struct 〈1, ;15〉 | |
12 | baseid 17209 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
13 | snsstp1 4816 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉} ⊆ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), (𝐶 Nat 𝐷)〉, 〈(comp‘ndx), (comp‘𝑄)〉} | |
14 | ovexd 7449 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) ∈ V) | |
15 | eqid 2726 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
16 | 10, 11, 12, 13, 14, 15 | strfv3 17200 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Base‘𝑄) = (𝐶 Func 𝐷)) |
17 | 16 | eqcomd 2732 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = (Base‘𝑄)) |
18 | base0 17211 | . . 3 ⊢ ∅ = (Base‘∅) | |
19 | funcrcl 17875 | . . . . 5 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
20 | 19 | con3i 154 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷)) |
21 | 20 | eq0rdv 4402 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅) |
22 | fnfuc 17961 | . . . . . . 7 ⊢ FuncCat Fn (Cat × Cat) | |
23 | 22 | fndmi 6654 | . . . . . 6 ⊢ dom FuncCat = (Cat × Cat) |
24 | 23 | ndmov 7600 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅) |
25 | 1, 24 | eqtrid 2778 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅) |
26 | 25 | fveq2d 6895 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Base‘𝑄) = (Base‘∅)) |
27 | 18, 21, 26 | 3eqtr4a 2792 | . 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = (Base‘𝑄)) |
28 | 17, 27 | pm2.61i 182 | 1 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3463 ∅c0 4323 {ctp 4628 〈cop 4630 × cxp 5671 ‘cfv 6544 (class class class)co 7414 1c1 11148 5c5 12314 ;cdc 12721 ndxcnx 17188 Basecbs 17206 Hom chom 17270 compcco 17271 Catccat 17670 Func cfunc 17866 Nat cnat 17957 FuncCat cfuc 17958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-fz 13531 df-struct 17142 df-slot 17177 df-ndx 17189 df-base 17207 df-hom 17283 df-cco 17284 df-func 17870 df-fuc 17960 |
This theorem is referenced by: fuccatid 17987 fucsect 17990 fucinv 17991 fuciso 17993 evlfcllem 18239 evlfcl 18240 curfcl 18250 uncf1 18254 uncf2 18255 curfuncf 18256 diag1cl 18260 curf2ndf 18265 yon1cl 18281 oyon1cl 18289 yonedalem21 18291 yonedalem22 18296 yonedalem3b 18297 yonedalem3 18298 yonedainv 18299 yonffthlem 18300 yoneda 18301 yoniso 18303 |
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