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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 2610 | . . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷) | |
3 | fuchom.n | . . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | eqid 2610 | . . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
6 | simpl 472 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat) | |
7 | simpr 476 | . . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat) | |
8 | eqid 2610 | . . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fuccofval 16442 | . . . . 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 16441 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉}) |
11 | catstr 16440 | . . . 4 ⊢ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} Struct 〈1, ;15〉 | |
12 | homid 15898 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
13 | snsstp2 4288 | . . . 4 ⊢ {〈(Hom ‘ndx), 𝑁〉} ⊆ {〈(Base‘ndx), (𝐶 Func 𝐷)〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (comp‘𝑄)〉} | |
14 | ovex 6577 | . . . . . 6 ⊢ (𝐶 Nat 𝐷) ∈ V | |
15 | 3, 14 | eqeltri 2684 | . . . . 5 ⊢ 𝑁 ∈ V |
16 | 15 | a1i 11 | . . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V) |
17 | eqid 2610 | . . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄) | |
18 | 10, 11, 12, 13, 16, 17 | strfv3 15736 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁) |
19 | 18 | eqcomd 2616 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
20 | df-hom 15793 | . . . 4 ⊢ Hom = Slot ;14 | |
21 | 20 | str0 15739 | . . 3 ⊢ ∅ = (Hom ‘∅) |
22 | 3 | natffn 16432 | . . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
23 | funcrcl 16346 | . . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
24 | 23 | con3i 149 | . . . . . . . . 9 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷)) |
25 | 24 | eq0rdv 3931 | . . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅) |
26 | 25 | xpeq2d 5063 | . . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅)) |
27 | xp0 5471 | . . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅ | |
28 | 26, 27 | syl6eq 2660 | . . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅) |
29 | 28 | fneq2d 5896 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ 𝑁 Fn ∅)) |
30 | 22, 29 | mpbii 222 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 Fn ∅) |
31 | fn0 5924 | . . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅) | |
32 | 30, 31 | sylib 207 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅) |
33 | fnfuc 16428 | . . . . . . 7 ⊢ FuncCat Fn (Cat × Cat) | |
34 | fndm 5904 | . . . . . . 7 ⊢ ( FuncCat Fn (Cat × Cat) → dom FuncCat = (Cat × Cat)) | |
35 | 33, 34 | ax-mp 5 | . . . . . 6 ⊢ dom FuncCat = (Cat × Cat) |
36 | 35 | ndmov 6716 | . . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅) |
37 | 1, 36 | syl5eq 2656 | . . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅) |
38 | 37 | fveq2d 6107 | . . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅)) |
39 | 21, 32, 38 | 3eqtr4a 2670 | . 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄)) |
40 | 19, 39 | pm2.61i 175 | 1 ⊢ 𝑁 = (Hom ‘𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {ctp 4129 〈cop 4131 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 1c1 9816 4c4 10949 5c5 10950 ;cdc 11369 ndxcnx 15692 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Func cfunc 16337 Nat cnat 16424 FuncCat cfuc 16425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-func 16341 df-nat 16426 df-fuc 16427 |
This theorem is referenced by: fuccatid 16452 fucsect 16455 fuciso 16458 evlfcllem 16684 evlfcl 16685 curfcl 16695 uncf2 16700 curfuncf 16701 diag2cl 16709 curf2ndf 16710 yonedalem21 16736 yonedalem22 16741 yonedalem3b 16742 yonedalem3 16743 yonffthlem 16745 |
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