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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 2818 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
5 | eqid 2818 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
6 | eqid 2818 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
7 | eqid 2818 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
8 | eqid 2818 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
9 | eqid 2818 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
10 | df-br 5058 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
11 | 1, 10 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
12 | funcrcl 17121 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
14 | 13 | simpld 495 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
15 | 13 | simprd 496 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) |
16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 17122 | . . 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 233 | . 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 1134 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 〈cop 4563 class class class wbr 5057 × 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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
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-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-ixp 8450 df-func 17116 |
This theorem is referenced by: funcsect 17130 funcinv 17131 funciso 17132 funcoppc 17133 cofu1 17142 cofucl 17146 cofuass 17147 cofulid 17148 cofurid 17149 funcres 17154 funcres2 17156 wunfunc 17157 funcres2c 17159 fullpropd 17178 fthsect 17183 fthinv 17184 fthmon 17185 ffthiso 17187 cofull 17192 cofth 17193 fuccocl 17222 fucidcl 17223 fuclid 17224 fucrid 17225 fucass 17226 fucsect 17230 fucinv 17231 invfuc 17232 fuciso 17233 natpropd 17234 fucpropd 17235 catciso 17355 prfval 17437 prfcl 17441 prf1st 17442 prf2nd 17443 1st2ndprf 17444 evlfcllem 17459 evlfcl 17460 curf1cl 17466 curfcl 17470 uncf1 17474 uncf2 17475 curfuncf 17476 uncfcurf 17477 diag1cl 17480 curf2ndf 17485 yon1cl 17501 oyon1cl 17509 yonedalem3a 17512 yonedalem4c 17515 yonedalem3b 17517 yonedalem3 17518 yonedainv 17519 yonffthlem 17520 yoniso 17523 |
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