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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.) (Proof shortened by AV, 14-Oct-2024.) |
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 2735 | . . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷) | |
3 | fuchom.n | . . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | eqid 2735 | . . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
6 | simpl 482 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat) | |
7 | simpr 484 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat) | |
8 | eqid 2735 | . . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fuccofval 18015 | . . . . 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 18014 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉}) |
11 | catstr 18013 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} Struct 〈1, ;15〉 | |
12 | homid 17458 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
13 | snsstp2 4822 | . . . 4 ⊢ {〈(Hom ‘ndx), 𝑁〉} ⊆ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} | |
14 | 3 | ovexi 7465 | . . . . 5 ⊢ 𝑁 ∈ V |
15 | 14 | a1i 11 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V) |
16 | eqid 2735 | . . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄) | |
17 | 10, 11, 12, 13, 15, 16 | strfv3 17239 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁) |
18 | 17 | eqcomd 2741 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
19 | 12 | str0 17223 | . . 3 ⊢ ∅ = (Hom ‘∅) |
20 | 3 | natffn 18004 | . . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
21 | funcrcl 17914 | . . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
22 | 21 | con3i 154 | . . . . . . . . 9 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷)) |
23 | 22 | eq0rdv 4413 | . . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅) |
24 | 23 | xpeq2d 5719 | . . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅)) |
25 | xp0 6180 | . . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅ | |
26 | 24, 25 | eqtrdi 2791 | . . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅) |
27 | 26 | fneq2d 6663 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ 𝑁 Fn ∅)) |
28 | 20, 27 | mpbii 233 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 Fn ∅) |
29 | fn0 6700 | . . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅) | |
30 | 28, 29 | sylib 218 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅) |
31 | fnfuc 18000 | . . . . . . 7 ⊢ FuncCat Fn (Cat × Cat) | |
32 | 31 | fndmi 6673 | . . . . . 6 ⊢ dom FuncCat = (Cat × Cat) |
33 | 32 | ndmov 7617 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅) |
34 | 1, 33 | eqtrid 2787 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅) |
35 | 34 | fveq2d 6911 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅)) |
36 | 19, 30, 35 | 3eqtr4a 2801 | . 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
37 | 18, 36 | pm2.61i 182 | 1 ⊢ 𝑁 = (Hom ‘𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {ctp 4635 〈cop 4637 × cxp 5687 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 1c1 11154 5c5 12322 ;cdc 12731 ndxcnx 17227 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Func cfunc 17905 Nat cnat 17996 FuncCat cfuc 17997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-hom 17322 df-cco 17323 df-func 17909 df-nat 17998 df-fuc 17999 |
This theorem is referenced by: fuccatid 18026 fucsect 18029 fuciso 18032 evlfcllem 18278 evlfcl 18279 curfcl 18289 uncf2 18294 curfuncf 18295 diag2cl 18303 curf2ndf 18304 yonedalem21 18330 yonedalem22 18335 yonedalem3b 18336 yonedalem3 18337 yonffthlem 18339 |
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