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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 2736 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
5 | eqid 2736 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
6 | eqid 2736 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
7 | eqid 2736 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
8 | eqid 2736 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
9 | eqid 2736 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
10 | df-br 5040 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
11 | 1, 10 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
12 | funcrcl 17323 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
14 | 13 | simpld 498 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
15 | 13 | simprd 499 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) |
16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 17324 | . . 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 235 | . 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 1144 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 〈cop 4533 class class class wbr 5039 × cxp 5534 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 ↑m cmap 8486 Xcixp 8556 Basecbs 16666 Hom chom 16760 compcco 16761 Catccat 17121 Idccid 17122 Func cfunc 17314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-ixp 8557 df-func 17318 |
This theorem is referenced by: funcsect 17332 funcinv 17333 funciso 17334 funcoppc 17335 cofu1 17344 cofucl 17348 cofuass 17349 cofulid 17350 cofurid 17351 funcres 17356 funcres2 17358 wunfunc 17359 wunfuncOLD 17360 funcres2c 17362 fullpropd 17381 fthsect 17386 fthinv 17387 fthmon 17388 ffthiso 17390 cofull 17395 cofth 17396 fuccocl 17427 fucidcl 17428 fuclid 17429 fucrid 17430 fucass 17431 fucsect 17435 fucinv 17436 invfuc 17437 fuciso 17438 natpropd 17439 fucpropd 17440 catciso 17571 prfval 17660 prfcl 17664 prf1st 17665 prf2nd 17666 1st2ndprf 17667 evlfcllem 17683 evlfcl 17684 curf1cl 17690 curfcl 17694 uncf1 17698 uncf2 17699 curfuncf 17700 uncfcurf 17701 diag1cl 17704 curf2ndf 17709 yon1cl 17725 oyon1cl 17733 yonedalem3a 17736 yonedalem4c 17739 yonedalem3b 17741 yonedalem3 17742 yonedainv 17743 yonffthlem 17744 yoniso 17747 fullthinc 45943 |
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