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Mirrors > Home > MPE Home > Th. List > relfunc | Structured version Visualization version GIF version |
Description: The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
relfunc | ⊢ Rel (𝐷 Func 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-func 17872 | . 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 | relmpoopab 8100 | 1 ⊢ Rel (𝐷 Func 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 [wsbc 3775 〈cop 4629 × cxp 5672 Rel wrel 5679 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 1st c1st 7993 2nd c2nd 7994 ↑m cmap 8847 Xcixp 8918 Basecbs 17208 Hom chom 17272 compcco 17273 Catccat 17672 Idccid 17673 Func cfunc 17868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-func 17872 |
This theorem is referenced by: cofuval 17896 cofu1 17898 cofu2 17900 cofuval2 17901 cofucl 17902 cofuass 17903 cofulid 17904 cofurid 17905 funcres 17910 funcres2 17912 wunfunc 17915 wunfuncOLD 17916 funcpropd 17917 relfull 17925 relfth 17926 isfull 17927 isfth 17931 idffth 17950 cofull 17951 cofth 17952 ressffth 17955 isnat 17965 isnat2 17966 nat1st2nd 17969 fuccocl 17984 fucidcl 17985 fuclid 17986 fucrid 17987 fucass 17988 fucsect 17992 fucinv 17993 invfuc 17994 fuciso 17995 natpropd 17996 fucpropd 17997 catciso 18128 prfval 18218 prfcl 18222 prf1st 18223 prf2nd 18224 1st2ndprf 18225 evlfcllem 18241 evlfcl 18242 curf1cl 18248 curf2cl 18251 curfcl 18252 uncf1 18256 uncf2 18257 curfuncf 18258 uncfcurf 18259 diag1cl 18262 diag2cl 18266 curf2ndf 18267 yon1cl 18283 oyon1cl 18291 yonedalem1 18292 yonedalem21 18293 yonedalem3a 18294 yonedalem4c 18297 yonedalem22 18298 yonedalem3b 18299 yonedalem3 18300 yonedainv 18301 yonffthlem 18302 yoniso 18305 |
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