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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 17808 | . 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 7648 | 1 ⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 [wsbc 3778 ⟨cop 4635 {copab 5211 × cxp 5675 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1st c1st 7973 2nd c2nd 7974 ↑m cmap 8820 Xcixp 8891 Basecbs 17144 Hom chom 17208 compcco 17209 Catccat 17608 Idccid 17609 Func cfunc 17804 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-dm 5687 df-iota 6496 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-func 17808 |
This theorem is referenced by: funcf1 17816 funcixp 17817 funcid 17820 funcco 17821 funcsect 17822 funcinv 17823 funciso 17824 funcoppc 17825 cofucl 17838 cofulid 17840 cofurid 17841 funcres 17846 funcres2b 17847 funcpropd 17851 funcres2c 17852 isfull 17861 isfth 17865 fthsect 17876 fthinv 17877 fthmon 17878 fthepi 17879 ffthiso 17880 natfval 17897 fucbas 17912 fuchom 17913 fuchomOLD 17914 fucco 17915 fuccocl 17917 fucidcl 17918 fuclid 17919 fucrid 17920 fucass 17921 fucid 17924 fucsect 17925 fucinv 17926 invfuc 17927 fuciso 17928 funcsetcres2 18043 prfcl 18155 prf1st 18156 prf2nd 18157 curf1cl 18181 curfcl 18185 uncfval 18187 uncfcl 18188 uncf1 18189 uncf2 18190 curfuncf 18191 uncfcurf 18192 yonffthlem 18235 yoneda 18236 |
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