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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 2818 | . . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷) | |
3 | fuchom.n | . . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | eqid 2818 | . . . . 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 2818 | . . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fuccofval 17217 | . . . . 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 17216 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉}) |
11 | catstr 17215 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} Struct 〈1, ;15〉 | |
12 | homid 16676 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
13 | snsstp2 4742 | . . . 4 ⊢ {〈(Hom ‘ndx), 𝑁〉} ⊆ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} | |
14 | 3 | ovexi 7179 | . . . . 5 ⊢ 𝑁 ∈ V |
15 | 14 | a1i 11 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V) |
16 | eqid 2818 | . . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄) | |
17 | 10, 11, 12, 13, 15, 16 | strfv3 16520 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁) |
18 | 17 | eqcomd 2824 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
19 | df-hom 16577 | . . . 4 ⊢ Hom = Slot ;14 | |
20 | 19 | str0 16523 | . . 3 ⊢ ∅ = (Hom ‘∅) |
21 | 3 | natffn 17207 | . . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
22 | funcrcl 17121 | . . . . . . . . . 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 4354 | . . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅) |
25 | 24 | xpeq2d 5578 | . . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅)) |
26 | xp0 6008 | . . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅ | |
27 | 25, 26 | syl6eq 2869 | . . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅) |
28 | 27 | fneq2d 6440 | . . . . 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 6472 | . . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅) | |
31 | 29, 30 | sylib 219 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅) |
32 | fnfuc 17203 | . . . . . . 7 ⊢ FuncCat Fn (Cat × Cat) | |
33 | fndm 6448 | . . . . . . 7 ⊢ ( FuncCat Fn (Cat × Cat) → dom FuncCat = (Cat × Cat)) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ dom FuncCat = (Cat × Cat) |
35 | 34 | ndmov 7321 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅) |
36 | 1, 35 | syl5eq 2865 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅) |
37 | 36 | fveq2d 6667 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅)) |
38 | 20, 31, 37 | 3eqtr4a 2879 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {ctp 4561 〈cop 4563 × cxp 5546 dom cdm 5548 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 1c1 10526 4c4 11682 5c5 11683 ;cdc 12086 ndxcnx 16468 Basecbs 16471 Hom chom 16564 compcco 16565 Catccat 16923 Func cfunc 17112 Nat cnat 17199 FuncCat cfuc 17200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-func 17116 df-nat 17201 df-fuc 17202 |
This theorem is referenced by: fuccatid 17227 fucsect 17230 fuciso 17233 evlfcllem 17459 evlfcl 17460 curfcl 17470 uncf2 17475 curfuncf 17476 diag2cl 17484 curf2ndf 17485 yonedalem21 17511 yonedalem22 17516 yonedalem3b 17517 yonedalem3 17518 yonffthlem 17520 |
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