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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 2821 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
5 | eqid 2821 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
6 | eqid 2821 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
7 | eqid 2821 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
8 | eqid 2821 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
9 | eqid 2821 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
10 | df-br 5067 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
11 | 1, 10 | sylib 220 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
12 | funcrcl 17133 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
14 | 13 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
15 | 13 | simprd 498 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) |
16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 17134 | . . 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 234 | . 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 1138 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 〈cop 4573 class class class wbr 5066 × cxp 5553 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 ↑m cmap 8406 Xcixp 8461 Basecbs 16483 Hom chom 16576 compcco 16577 Catccat 16935 Idccid 16936 Func cfunc 17124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-ixp 8462 df-func 17128 |
This theorem is referenced by: funcsect 17142 funcinv 17143 funciso 17144 funcoppc 17145 cofu1 17154 cofucl 17158 cofuass 17159 cofulid 17160 cofurid 17161 funcres 17166 funcres2 17168 wunfunc 17169 funcres2c 17171 fullpropd 17190 fthsect 17195 fthinv 17196 fthmon 17197 ffthiso 17199 cofull 17204 cofth 17205 fuccocl 17234 fucidcl 17235 fuclid 17236 fucrid 17237 fucass 17238 fucsect 17242 fucinv 17243 invfuc 17244 fuciso 17245 natpropd 17246 fucpropd 17247 catciso 17367 prfval 17449 prfcl 17453 prf1st 17454 prf2nd 17455 1st2ndprf 17456 evlfcllem 17471 evlfcl 17472 curf1cl 17478 curfcl 17482 uncf1 17486 uncf2 17487 curfuncf 17488 uncfcurf 17489 diag1cl 17492 curf2ndf 17497 yon1cl 17513 oyon1cl 17521 yonedalem3a 17524 yonedalem4c 17527 yonedalem3b 17529 yonedalem3 17530 yonedainv 17531 yonffthlem 17532 yoniso 17535 |
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