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Mirrors > Home > MPE Home > Th. List > funcsetcestrc | Structured version Visualization version GIF version |
Description: The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcsetcestrc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcsetcestrc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
funcsetcestrc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcsetcestrc.o | ⊢ (𝜑 → ω ∈ 𝑈) |
funcsetcestrc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) |
funcsetcestrc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
Ref | Expression |
---|---|
funcsetcestrc | ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.c | . 2 ⊢ 𝐶 = (Base‘𝑆) | |
2 | eqid 2821 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
3 | eqid 2821 | . 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆) | |
4 | eqid 2821 | . 2 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
5 | eqid 2821 | . 2 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
6 | eqid 2821 | . 2 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
7 | eqid 2821 | . 2 ⊢ (comp‘𝑆) = (comp‘𝑆) | |
8 | eqid 2821 | . 2 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
9 | funcsetcestrc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
10 | funcsetcestrc.s | . . . 4 ⊢ 𝑆 = (SetCat‘𝑈) | |
11 | 10 | setccat 17339 | . . 3 ⊢ (𝑈 ∈ WUni → 𝑆 ∈ Cat) |
12 | 9, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ∈ Cat) |
13 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
14 | 13 | estrccat 17377 | . . 3 ⊢ (𝑈 ∈ WUni → 𝐸 ∈ Cat) |
15 | 9, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝐸 ∈ Cat) |
16 | funcsetcestrc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
17 | funcsetcestrc.o | . . 3 ⊢ (𝜑 → ω ∈ 𝑈) | |
18 | 10, 1, 16, 9, 17, 13, 2 | funcsetcestrclem3 17400 | . 2 ⊢ (𝜑 → 𝐹:𝐶⟶(Base‘𝐸)) |
19 | funcsetcestrc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) | |
20 | 10, 1, 16, 9, 17, 19 | funcsetcestrclem4 17402 | . 2 ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶)) |
21 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem8 17406 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹‘𝑎)(Hom ‘𝐸)(𝐹‘𝑏))) |
22 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem7 17405 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ((𝑎𝐺𝑎)‘((Id‘𝑆)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎))) |
23 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem9 17407 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ∧ 𝑐 ∈ 𝐶) ∧ (ℎ ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑏(Hom ‘𝑆)𝑐))) → ((𝑎𝐺𝑐)‘(𝑘(〈𝑎, 𝑏〉(comp‘𝑆)𝑐)ℎ)) = (((𝑏𝐺𝑐)‘𝑘)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘ℎ))) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 18, 20, 21, 22, 23 | isfuncd 17129 | 1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4560 〈cop 4566 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ωcom 7574 ↑m cmap 8400 WUnicwun 10116 ndxcnx 16474 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Idccid 16930 Func cfunc 17118 SetCatcsetc 17329 ExtStrCatcestrc 17366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-wun 10118 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-c 10537 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-hom 16583 df-cco 16584 df-cat 16933 df-cid 16934 df-func 17122 df-setc 17330 df-estrc 17367 |
This theorem is referenced by: fthsetcestrc 17409 fullsetcestrc 17410 |
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