Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17116 | . 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 7376 | 1 ⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 [wsbc 3769 〈cop 4563 {copab 5119 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 ↑m cmap 8395 Xcixp 8449 Basecbs 16471 Hom chom 16564 compcco 16565 Catccat 16923 Idccid 16924 Func cfunc 17112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-func 17116 |
This theorem is referenced by: funcf1 17124 funcixp 17125 funcid 17128 funcco 17129 funcsect 17130 funcinv 17131 funciso 17132 funcoppc 17133 cofucl 17146 cofulid 17148 cofurid 17149 funcres 17154 funcres2b 17155 funcpropd 17158 funcres2c 17159 isfull 17168 isfth 17172 fthsect 17183 fthinv 17184 fthmon 17185 fthepi 17186 ffthiso 17187 natfval 17204 fucbas 17218 fuchom 17219 fucco 17220 fuccocl 17222 fucidcl 17223 fuclid 17224 fucrid 17225 fucass 17226 fucid 17229 fucsect 17230 fucinv 17231 invfuc 17232 fuciso 17233 funcsetcres2 17341 prfcl 17441 prf1st 17442 prf2nd 17443 curf1cl 17466 curfcl 17470 uncfval 17472 uncfcl 17473 uncf1 17474 uncf2 17475 curfuncf 17476 uncfcurf 17477 yonffthlem 17520 yoneda 17521 |
Copyright terms: Public domain | W3C validator |