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Mirrors > Home > MPE Home > Th. List > equivestrcsetc | Structured version Visualization version GIF version |
Description: The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (Contributed by AV, 2-Apr-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcestrcsetc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcestrcsetc.b | ⊢ 𝐵 = (Base‘𝐸) |
funcestrcsetc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcestrcsetc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcestrcsetc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcestrcsetc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) |
equivestrcsetc.i | ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) |
Ref | Expression |
---|---|
equivestrcsetc | ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.e | . . 3 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
2 | funcestrcsetc.s | . . 3 ⊢ 𝑆 = (SetCat‘𝑈) | |
3 | funcestrcsetc.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
4 | funcestrcsetc.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
5 | funcestrcsetc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcestrcsetc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) | |
7 | funcestrcsetc.g | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fthestrcsetc 17392 | . 2 ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺) |
9 | 1, 2, 3, 4, 5, 6, 7 | fullestrcsetc 17393 | . 2 ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺) |
10 | 2, 5 | setcbas 17330 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
11 | 4, 10 | eqtr4id 2852 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝑈) |
12 | 11 | eleq2d 2875 | . . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ 𝑈)) |
13 | eqid 2798 | . . . . . . . . 9 ⊢ {〈(Base‘ndx), 𝑏〉} = {〈(Base‘ndx), 𝑏〉} | |
14 | equivestrcsetc.i | . . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) | |
15 | 13, 5, 14 | 1strwunbndx 16592 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑈) → {〈(Base‘ndx), 𝑏〉} ∈ 𝑈) |
16 | 15 | ex 416 | . . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝑈 → {〈(Base‘ndx), 𝑏〉} ∈ 𝑈)) |
17 | 12, 16 | sylbid 243 | . . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝐶 → {〈(Base‘ndx), 𝑏〉} ∈ 𝑈)) |
18 | 17 | imp 410 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {〈(Base‘ndx), 𝑏〉} ∈ 𝑈) |
19 | 1, 5 | estrcbas 17367 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐸)) |
20 | 19 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑈 = (Base‘𝐸)) |
21 | 3, 20 | eqtr4id 2852 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝐵 = 𝑈) |
22 | 18, 21 | eleqtrrd 2893 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → {〈(Base‘ndx), 𝑏〉} ∈ 𝐵) |
23 | fveq2 6645 | . . . . . . 7 ⊢ (𝑎 = {〈(Base‘ndx), 𝑏〉} → (𝐹‘𝑎) = (𝐹‘{〈(Base‘ndx), 𝑏〉})) | |
24 | 23 | f1oeq3d 6587 | . . . . . 6 ⊢ (𝑎 = {〈(Base‘ndx), 𝑏〉} → (𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ 𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}))) |
25 | 24 | exbidv 1922 | . . . . 5 ⊢ (𝑎 = {〈(Base‘ndx), 𝑏〉} → (∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}))) |
26 | 25 | adantl 485 | . . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑎 = {〈(Base‘ndx), 𝑏〉}) → (∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎) ↔ ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}))) |
27 | f1oi 6627 | . . . . . 6 ⊢ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏 | |
28 | 1, 2, 3, 4, 5, 6 | funcestrcsetclem1 17382 | . . . . . . . . 9 ⊢ ((𝜑 ∧ {〈(Base‘ndx), 𝑏〉} ∈ 𝐵) → (𝐹‘{〈(Base‘ndx), 𝑏〉}) = (Base‘{〈(Base‘ndx), 𝑏〉})) |
29 | 22, 28 | syldan 594 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{〈(Base‘ndx), 𝑏〉}) = (Base‘{〈(Base‘ndx), 𝑏〉})) |
30 | 13 | 1strbas 16591 | . . . . . . . . 9 ⊢ (𝑏 ∈ 𝐶 → 𝑏 = (Base‘{〈(Base‘ndx), 𝑏〉})) |
31 | 30 | adantl 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑏 = (Base‘{〈(Base‘ndx), 𝑏〉})) |
32 | 29, 31 | eqtr4d 2836 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (𝐹‘{〈(Base‘ndx), 𝑏〉}) = 𝑏) |
33 | 32 | f1oeq3d 6587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → (( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}) ↔ ( I ↾ 𝑏):𝑏–1-1-onto→𝑏)) |
34 | 27, 33 | mpbiri 261 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉})) |
35 | resiexg 7601 | . . . . . . 7 ⊢ (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V) | |
36 | 35 | elv 3446 | . . . . . 6 ⊢ ( I ↾ 𝑏) ∈ V |
37 | f1oeq1 6579 | . . . . . 6 ⊢ (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}) ↔ ( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}))) | |
38 | 36, 37 | spcev 3555 | . . . . 5 ⊢ (( I ↾ 𝑏):𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉}) → ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉})) |
39 | 34, 38 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘{〈(Base‘ndx), 𝑏〉})) |
40 | 22, 26, 39 | rspcedvd 3574 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎)) |
41 | 40 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎)) |
42 | 8, 9, 41 | 3jca 1125 | 1 ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 {csn 4525 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑m cmap 8389 WUnicwun 10111 ndxcnx 16472 Basecbs 16475 Full cful 17164 Faith cfth 17165 SetCatcsetc 17327 ExtStrCatcestrc 17364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wun 10113 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-hom 16581 df-cco 16582 df-cat 16931 df-cid 16932 df-func 17120 df-full 17166 df-fth 17167 df-setc 17328 df-estrc 17365 |
This theorem is referenced by: (None) |
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