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| Mirrors > Home > MPE Home > Th. List > funcf1 | Structured version Visualization version GIF version | ||
| Description: The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| funcf1.b | ⊢ 𝐵 = (Base‘𝐷) |
| funcf1.c | ⊢ 𝐶 = (Base‘𝐸) |
| funcf1.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
| Ref | Expression |
|---|---|
| funcf1 | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funcf1.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
| 2 | funcf1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
| 3 | funcf1.c | . . . 4 ⊢ 𝐶 = (Base‘𝐸) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
| 5 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
| 6 | eqid 2737 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
| 7 | eqid 2737 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
| 8 | eqid 2737 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
| 9 | eqid 2737 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
| 10 | df-br 5144 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
| 11 | 1, 10 | sylib 218 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
| 12 | funcrcl 17908 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
| 13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
| 14 | 13 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
| 15 | 13 | simprd 495 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) |
| 16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 17909 | . . 3 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))) |
| 17 | 1, 16 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))) |
| 18 | 17 | simp1d 1143 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 class class class wbr 5143 × 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-ixp 8938 df-func 17903 |
| This theorem is referenced by: funcsect 17917 funcinv 17918 funciso 17919 funcoppc 17920 cofu1 17929 cofucl 17933 cofuass 17934 cofulid 17935 cofurid 17936 funcres 17941 funcres2 17943 wunfunc 17946 funcres2c 17948 fullpropd 17967 fthsect 17972 fthinv 17973 fthmon 17974 ffthiso 17976 cofull 17981 cofth 17982 fuccocl 18012 fucidcl 18013 fuclid 18014 fucrid 18015 fucass 18016 fucsect 18020 fucinv 18021 invfuc 18022 fuciso 18023 natpropd 18024 fucpropd 18025 catciso 18156 prfval 18244 prfcl 18248 prf1st 18249 prf2nd 18250 1st2ndprf 18251 evlfcllem 18266 evlfcl 18267 curf1cl 18273 curfcl 18277 uncf1 18281 uncf2 18282 curfuncf 18283 uncfcurf 18284 diag1cl 18287 curf2ndf 18292 yon1cl 18308 oyon1cl 18316 yonedalem3a 18319 yonedalem4c 18322 yonedalem3b 18324 yonedalem3 18325 yonedainv 18326 yonffthlem 18327 yoniso 18330 upciclem2 48924 upciclem3 48925 upeu2 48929 oppcup 48948 diag1 49004 fuco111x 49026 fuco11idx 49030 fuco22natlem1 49037 fuco22natlem2 49038 fuco22natlem3 49039 fuco22natlem 49040 fucoid 49043 fuco23alem 49046 fucocolem1 49048 fucocolem2 49049 fucocolem3 49050 fucocolem4 49051 fucoco 49052 fucolid 49056 fucorid 49057 fucorid2 49058 precofvallem 49061 precofvalALT 49063 precofval2 49064 functhincfun 49098 fullthinc 49099 thincciso2 49104 functermc 49140 fulltermc 49143 |
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