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Mirrors > Home > MPE Home > Th. List > fuchom | Structured version Visualization version GIF version |
Description: The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
fucbas.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
fuchom.n | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
Ref | Expression |
---|---|
fuchom | ⊢ 𝑁 = (Hom ‘𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucbas.q | . . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
2 | eqid 2794 | . . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷) | |
3 | fuchom.n | . . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | eqid 2794 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | eqid 2794 | . . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
6 | simpl 483 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat) | |
7 | simpr 485 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat) | |
8 | eqid 2794 | . . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fuccofval 17058 | . . . . 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 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (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 17057 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉}) |
11 | catstr 17056 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} Struct 〈1, ;15〉 | |
12 | homid 16517 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
13 | snsstp2 4659 | . . . 4 ⊢ {〈(Hom ‘ndx), 𝑁〉} ⊆ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} | |
14 | 3 | ovexi 7052 | . . . . 5 ⊢ 𝑁 ∈ V |
15 | 14 | a1i 11 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V) |
16 | eqid 2794 | . . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄) | |
17 | 10, 11, 12, 13, 15, 16 | strfv3 16361 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁) |
18 | 17 | eqcomd 2800 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
19 | df-hom 16418 | . . . 4 ⊢ Hom = Slot ;14 | |
20 | 19 | str0 16364 | . . 3 ⊢ ∅ = (Hom ‘∅) |
21 | 3 | natffn 17048 | . . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
22 | funcrcl 16962 | . . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
23 | 22 | con3i 157 | . . . . . . . . 9 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷)) |
24 | 23 | eq0rdv 4279 | . . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅) |
25 | 24 | xpeq2d 5476 | . . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅)) |
26 | xp0 5894 | . . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅ | |
27 | 25, 26 | syl6eq 2846 | . . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅) |
28 | 27 | fneq2d 6320 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ 𝑁 Fn ∅)) |
29 | 21, 28 | mpbii 234 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 Fn ∅) |
30 | fn0 6350 | . . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅) | |
31 | 29, 30 | sylib 219 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅) |
32 | fnfuc 17044 | . . . . . . 7 ⊢ FuncCat Fn (Cat × Cat) | |
33 | fndm 6328 | . . . . . . 7 ⊢ ( FuncCat Fn (Cat × Cat) → dom FuncCat = (Cat × Cat)) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ dom FuncCat = (Cat × Cat) |
35 | 34 | ndmov 7191 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅) |
36 | 1, 35 | syl5eq 2842 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅) |
37 | 36 | fveq2d 6545 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅)) |
38 | 20, 31, 37 | 3eqtr4a 2856 | . 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
39 | 18, 38 | pm2.61i 183 | 1 ⊢ 𝑁 = (Hom ‘𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1522 ∈ wcel 2080 Vcvv 3436 ∅c0 4213 {ctp 4478 〈cop 4480 × cxp 5444 dom cdm 5446 Fn wfn 6223 ‘cfv 6228 (class class class)co 7019 1c1 10387 4c4 11544 5c5 11545 ;cdc 11948 ndxcnx 16309 Basecbs 16312 Hom chom 16405 compcco 16406 Catccat 16764 Func cfunc 16953 Nat cnat 17040 FuncCat cfuc 17041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-ixp 8314 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-dec 11949 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-hom 16418 df-cco 16419 df-func 16957 df-nat 17042 df-fuc 17043 |
This theorem is referenced by: fuccatid 17068 fucsect 17071 fuciso 17074 evlfcllem 17300 evlfcl 17301 curfcl 17311 uncf2 17316 curfuncf 17317 diag2cl 17325 curf2ndf 17326 yonedalem21 17352 yonedalem22 17357 yonedalem3b 17358 yonedalem3 17359 yonffthlem 17361 |
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