| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > diagcl | Structured version Visualization version GIF version | ||
| Description: The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor (𝑦 ∈ 𝐷 ↦ 𝑋) is a construction that is natural in 𝑋 (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| Ref | Expression |
|---|---|
| diagval.l | ⊢ 𝐿 = (𝐶Δfunc𝐷) |
| diagval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| diagval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
| diagcl.q | ⊢ 𝑄 = (𝐷 FuncCat 𝐶) |
| Ref | Expression |
|---|---|
| diagcl | ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | diagval.l | . . 3 ⊢ 𝐿 = (𝐶Δfunc𝐷) | |
| 2 | diagval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 3 | diagval.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
| 4 | 1, 2, 3 | diagval 18285 | . 2 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
| 5 | eqid 2737 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
| 6 | diagcl.q | . . 3 ⊢ 𝑄 = (𝐷 FuncCat 𝐶) | |
| 7 | eqid 2737 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
| 8 | eqid 2737 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
| 9 | 7, 2, 3, 8 | 1stfcl 18242 | . . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
| 10 | 5, 6, 2, 3, 9 | curfcl 18277 | . 2 ⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) ∈ (𝐶 Func 𝑄)) |
| 11 | 4, 10 | eqeltrd 2841 | 1 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 (class class class)co 7431 Catccat 17707 Func cfunc 17899 FuncCat cfuc 17990 ×c cxpc 18213 1stF c1stf 18214 curryF ccurf 18255 Δfunccdiag 18257 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-cat 17711 df-cid 17712 df-func 17903 df-nat 17991 df-fuc 17992 df-xpc 18217 df-1stf 18218 df-curf 18259 df-diag 18261 |
| This theorem is referenced by: diag1cl 18287 diag2cl 18291 |
| Copyright terms: Public domain | W3C validator |