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Theorem equivestrcsetc 17394
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 17392 . 2 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7fullestrcsetc 17393 . 2 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
102, 5setcbas 17330 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
1110, 4syl6reqr 2873 . . . . . . . 8 (𝜑𝐶 = 𝑈)
1211eleq2d 2896 . . . . . . 7 (𝜑 → (𝑏𝐶𝑏𝑈))
13 eqid 2819 . . . . . . . . 9 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14 equivestrcsetc.i . . . . . . . . 9 (𝜑 → (Base‘ndx) ∈ 𝑈)
1513, 5, 141strwunbndx 16592 . . . . . . . 8 ((𝜑𝑏𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
1615ex 415 . . . . . . 7 (𝜑 → (𝑏𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1712, 16sylbid 242 . . . . . 6 (𝜑 → (𝑏𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1817imp 409 . . . . 5 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
191, 5estrcbas 17367 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
2019adantr 483 . . . . . 6 ((𝜑𝑏𝐶) → 𝑈 = (Base‘𝐸))
2120, 3syl6reqr 2873 . . . . 5 ((𝜑𝑏𝐶) → 𝐵 = 𝑈)
2218, 21eleqtrrd 2914 . . . 4 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23 fveq2 6663 . . . . . . 7 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
2423f1oeq3d 6605 . . . . . 6 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2524exbidv 1916 . . . . 5 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2625adantl 484 . . . 4 (((𝜑𝑏𝐶) ∧ 𝑎 = {⟨(Base‘ndx), 𝑏⟩}) → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
27 f1oi 6645 . . . . . 6 ( I ↾ 𝑏):𝑏1-1-onto𝑏
281, 2, 3, 4, 5, 6funcestrcsetclem1 17382 . . . . . . . . 9 ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
2922, 28syldan 593 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30131strbas 16591 . . . . . . . . 9 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130adantl 484 . . . . . . . 8 ((𝜑𝑏𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3229, 31eqtr4d 2857 . . . . . . 7 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
3332f1oeq3d 6605 . . . . . 6 ((𝜑𝑏𝐶) → (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto𝑏))
3427, 33mpbiri 260 . . . . 5 ((𝜑𝑏𝐶) → ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35 resiexg 7611 . . . . . . 7 (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
3635elv 3498 . . . . . 6 ( I ↾ 𝑏) ∈ V
37 f1oeq1 6597 . . . . . 6 (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
3836, 37spcev 3605 . . . . 5 (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
3934, 38syl 17 . . . 4 ((𝜑𝑏𝐶) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
4022, 26, 39rspcedvd 3624 . . 3 ((𝜑𝑏𝐶) → ∃𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
4140ralrimiva 3180 . 2 (𝜑 → ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
428, 9, 413jca 1123 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 208  wa 398  w3a 1082   = wceq 1531  wex 1774  wcel 2108  wral 3136  wrex 3137  Vcvv 3493  {csn 4559  cop 4565   class class class wbr 5057  cmpt 5137   I cid 5452  cres 5550  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  cmpo 7150  m cmap 8398  WUnicwun 10114  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 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-wun 10116  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  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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