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Theorem equivestrcsetc 17058
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.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b 𝐵 = (Base‘𝐸)
funcestrcsetc.c 𝐶 = (Base‘𝑆)
funcestrcsetc.u (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
equivestrcsetc.i (𝜑 → (Base‘ndx) ∈ 𝑈)
Assertion
Ref Expression
equivestrcsetc (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎)))
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝜑,𝑦   𝑎,𝑏,𝑥,𝑦,𝐵   𝐹,𝑎,𝑏   𝐺,𝑎,𝑏   𝐸,𝑎,𝑏   𝑆,𝑎,𝑏   𝜑,𝑎,𝑏   𝐶,𝑎   𝑖,𝐹,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑖)   𝐵(𝑖)   𝐶(𝑦,𝑖,𝑏)   𝑆(𝑥,𝑦,𝑖)   𝑈(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐸(𝑥,𝑦,𝑖)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑖)

Proof of Theorem equivestrcsetc
StepHypRef 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 (𝜑𝑈 ∈ WUni)
6 funcestrcsetc.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcestrcsetc.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
81, 2, 3, 4, 5, 6, 7fthestrcsetc 17056 . 2 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7fullestrcsetc 17057 . 2 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
102, 5setcbas 16993 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
1110, 4syl6reqr 2818 . . . . . . . 8 (𝜑𝐶 = 𝑈)
1211eleq2d 2830 . . . . . . 7 (𝜑 → (𝑏𝐶𝑏𝑈))
13 eqid 2765 . . . . . . . . 9 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14 equivestrcsetc.i . . . . . . . . 9 (𝜑 → (Base‘ndx) ∈ 𝑈)
1513, 5, 141strwunbndx 16253 . . . . . . . 8 ((𝜑𝑏𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
1615ex 401 . . . . . . 7 (𝜑 → (𝑏𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1712, 16sylbid 231 . . . . . 6 (𝜑 → (𝑏𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1817imp 395 . . . . 5 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
191, 5estrcbas 17030 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
2019adantr 472 . . . . . 6 ((𝜑𝑏𝐶) → 𝑈 = (Base‘𝐸))
2120, 3syl6reqr 2818 . . . . 5 ((𝜑𝑏𝐶) → 𝐵 = 𝑈)
2218, 21eleqtrrd 2847 . . . 4 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23 fveq2 6375 . . . . . . 7 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
2423f1oeq3d 6317 . . . . . 6 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2524exbidv 2016 . . . . 5 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2625adantl 473 . . . 4 (((𝜑𝑏𝐶) ∧ 𝑎 = {⟨(Base‘ndx), 𝑏⟩}) → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
27 f1oi 6357 . . . . . 6 ( I ↾ 𝑏):𝑏1-1-onto𝑏
281, 2, 3, 4, 5, 6funcestrcsetclem1 17046 . . . . . . . . 9 ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
2922, 28syldan 585 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30131strbas 16252 . . . . . . . . 9 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130adantl 473 . . . . . . . 8 ((𝜑𝑏𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3229, 31eqtr4d 2802 . . . . . . 7 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
3332f1oeq3d 6317 . . . . . 6 ((𝜑𝑏𝐶) → (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto𝑏))
3427, 33mpbiri 249 . . . . 5 ((𝜑𝑏𝐶) → ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35 vex 3353 . . . . . . 7 𝑏 ∈ V
36 resiexg 7300 . . . . . . 7 (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
3735, 36ax-mp 5 . . . . . 6 ( I ↾ 𝑏) ∈ V
38 f1oeq1 6310 . . . . . 6 (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
3937, 38spcev 3452 . . . . 5 (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
4034, 39syl 17 . . . 4 ((𝜑𝑏𝐶) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
4122, 26, 40rspcedvd 3468 . . 3 ((𝜑𝑏𝐶) → ∃𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
4241ralrimiva 3113 . 2 (𝜑 → ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
438, 9, 423jca 1158 1 (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  {csn 4334  cop 4340   class class class wbr 4809  cmpt 4888   I cid 5184  cres 5279  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑚 cmap 8060  WUnicwun 9775  ndxcnx 16127  Basecbs 16130   Full cful 16827   Faith cfth 16828  SetCatcsetc 16990  ExtStrCatcestrc 17027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-wun 9777  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-hom 16238  df-cco 16239  df-cat 16594  df-cid 16595  df-func 16783  df-full 16829  df-fth 16830  df-setc 16991  df-estrc 17028
This theorem is referenced by: (None)
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