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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 2734 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
3 | eqid 2734 | . 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆) | |
4 | eqid 2734 | . 2 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
5 | eqid 2734 | . 2 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
6 | eqid 2734 | . 2 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
7 | eqid 2734 | . 2 ⊢ (comp‘𝑆) = (comp‘𝑆) | |
8 | eqid 2734 | . 2 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
9 | funcsetcestrc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
10 | funcsetcestrc.s | . . . 4 ⊢ 𝑆 = (SetCat‘𝑈) | |
11 | 10 | setccat 18138 | . . 3 ⊢ (𝑈 ∈ WUni → 𝑆 ∈ Cat) |
12 | 9, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ∈ Cat) |
13 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
14 | 13 | estrccat 18187 | . . 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 18211 | . 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 18213 | . 2 ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶)) |
21 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem8 18217 | . 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 18216 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ((𝑎𝐺𝑎)‘((Id‘𝑆)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎))) |
23 | 10, 1, 16, 9, 17, 19, 13 | funcsetcestrclem9 18218 | . 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 17915 | 1 ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {csn 4630 〈cop 4636 class class class wbr 5147 ↦ cmpt 5230 I cid 5581 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ωcom 7886 ↑m cmap 8864 WUnicwun 10737 ndxcnx 17226 Basecbs 17244 Hom chom 17308 compcco 17309 Catccat 17708 Idccid 17709 Func cfunc 17904 SetCatcsetc 18128 ExtStrCatcestrc 18176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8743 df-ec 8745 df-qs 8749 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-wun 10739 df-ni 10909 df-pli 10910 df-mi 10911 df-lti 10912 df-plpq 10945 df-mpq 10946 df-ltpq 10947 df-enq 10948 df-nq 10949 df-erq 10950 df-plq 10951 df-mq 10952 df-1nq 10953 df-rq 10954 df-ltnq 10955 df-np 11018 df-plp 11020 df-ltp 11022 df-enr 11092 df-nr 11093 df-c 11158 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-hom 17321 df-cco 17322 df-cat 17712 df-cid 17713 df-func 17908 df-setc 18129 df-estrc 18177 |
This theorem is referenced by: fthsetcestrc 18220 fullsetcestrc 18221 |
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