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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 18134 | . 2 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))) |
5 | eqid 2733 | . . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) | |
6 | diagcl.q | . . 3 ⊢ 𝑄 = (𝐷 FuncCat 𝐶) | |
7 | eqid 2733 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
8 | eqid 2733 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
9 | 7, 2, 3, 8 | 1stfcl 18090 | . . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
10 | 5, 6, 2, 3, 9 | curfcl 18126 | . 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) ∈ (𝐶 Func 𝑄)) |
11 | 4, 10 | eqeltrd 2834 | 1 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 (class class class)co 7358 Catccat 17549 Func cfunc 17745 FuncCat cfuc 17834 ×c cxpc 18061 1stF c1stf 18062 curryF ccurf 18104 Δfunccdiag 18106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-func 17749 df-nat 17835 df-fuc 17836 df-xpc 18065 df-1stf 18066 df-curf 18108 df-diag 18110 |
This theorem is referenced by: diag1cl 18136 diag2cl 18140 |
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