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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 17909 | . 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 8118 | 1 ⊢ Rel (𝐷 Func 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 [wsbc 3791 〈cop 4637 × cxp 5687 Rel wrel 5694 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ↑m cmap 8865 Xcixp 8936 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 Func cfunc 17905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-func 17909 |
This theorem is referenced by: cofuval 17933 cofu1 17935 cofu2 17937 cofuval2 17938 cofucl 17939 cofuass 17940 cofulid 17941 cofurid 17942 funcres 17947 funcres2 17949 wunfunc 17952 wunfuncOLD 17953 funcpropd 17954 relfull 17962 relfth 17963 isfull 17964 isfth 17968 idffth 17987 cofull 17988 cofth 17989 ressffth 17992 isnat 18002 isnat2 18003 nat1st2nd 18006 fuccocl 18021 fucidcl 18022 fuclid 18023 fucrid 18024 fucass 18025 fucsect 18029 fucinv 18030 invfuc 18031 fuciso 18032 natpropd 18033 fucpropd 18034 catciso 18165 prfval 18255 prfcl 18259 prf1st 18260 prf2nd 18261 1st2ndprf 18262 evlfcllem 18278 evlfcl 18279 curf1cl 18285 curf2cl 18288 curfcl 18289 uncf1 18293 uncf2 18294 curfuncf 18295 uncfcurf 18296 diag1cl 18299 diag2cl 18303 curf2ndf 18304 yon1cl 18320 oyon1cl 18328 yonedalem1 18329 yonedalem21 18330 yonedalem3a 18331 yonedalem4c 18334 yonedalem22 18335 yonedalem3b 18336 yonedalem3 18337 yonedainv 18338 yonffthlem 18339 yoniso 18342 |
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