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| Mirrors > Home > MPE Home > Th. List > funcrcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| Ref | Expression |
|---|---|
| funcrcl | ⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-func 17903 | . 2 ⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {〈𝑓, 𝑔〉 ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(〈(𝑓‘𝑥), (𝑓‘𝑦)〉(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}) | |
| 2 | 1 | elmpocl 7674 | 1 ⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 〈cop 4632 {copab 5205 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 ↑m cmap 8866 Xcixp 8937 Basecbs 17247 Hom chom 17308 compcco 17309 Catccat 17707 Idccid 17708 Func cfunc 17899 |
| 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-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-func 17903 |
| This theorem is referenced by: funcf1 17911 funcixp 17912 funcid 17915 funcco 17916 funcsect 17917 funcinv 17918 funciso 17919 funcoppc 17920 cofucl 17933 cofulid 17935 cofurid 17936 funcres 17941 funcres2b 17942 funcpropd 17947 funcres2c 17948 isfull 17957 isfth 17961 fthsect 17972 fthinv 17973 fthmon 17974 fthepi 17975 ffthiso 17976 natfval 17994 fucbas 18008 fuchom 18009 fucco 18010 fuccocl 18012 fucidcl 18013 fuclid 18014 fucrid 18015 fucass 18016 fucid 18019 fucsect 18020 fucinv 18021 invfuc 18022 fuciso 18023 funcsetcres2 18138 prfcl 18248 prf1st 18249 prf2nd 18250 curf1cl 18273 curfcl 18277 uncfval 18279 uncfcl 18280 uncf1 18281 uncf2 18282 curfuncf 18283 uncfcurf 18284 yonffthlem 18327 yoneda 18328 funcrcl2 48912 funcrcl3 48913 |
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