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Theorem equivestrcsetc 18208
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 18206 . 2 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7fullestrcsetc 18207 . 2 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
102, 5setcbas 18132 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
114, 10eqtr4id 2794 . . . . . . . 8 (𝜑𝐶 = 𝑈)
1211eleq2d 2825 . . . . . . 7 (𝜑 → (𝑏𝐶𝑏𝑈))
13 eqid 2735 . . . . . . . . 9 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14 equivestrcsetc.i . . . . . . . . 9 (𝜑 → (Base‘ndx) ∈ 𝑈)
1513, 5, 141strwunbndx 17264 . . . . . . . 8 ((𝜑𝑏𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
1615ex 412 . . . . . . 7 (𝜑 → (𝑏𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1712, 16sylbid 240 . . . . . 6 (𝜑 → (𝑏𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1817imp 406 . . . . 5 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
191, 5estrcbas 18180 . . . . . . 7 (𝜑𝑈 = (Base‘𝐸))
2019adantr 480 . . . . . 6 ((𝜑𝑏𝐶) → 𝑈 = (Base‘𝐸))
213, 20eqtr4id 2794 . . . . 5 ((𝜑𝑏𝐶) → 𝐵 = 𝑈)
2218, 21eleqtrrd 2842 . . . 4 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
23 fveq2 6907 . . . . . . 7 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
2423f1oeq3d 6846 . . . . . 6 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2524exbidv 1919 . . . . 5 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2625adantl 481 . . . 4 (((𝜑𝑏𝐶) ∧ 𝑎 = {⟨(Base‘ndx), 𝑏⟩}) → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
27 f1oi 6887 . . . . . 6 ( I ↾ 𝑏):𝑏1-1-onto𝑏
281, 2, 3, 4, 5, 6funcestrcsetclem1 18196 . . . . . . . . 9 ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
2922, 28syldan 591 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
30131strbas 17262 . . . . . . . . 9 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130adantl 481 . . . . . . . 8 ((𝜑𝑏𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3229, 31eqtr4d 2778 . . . . . . 7 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
3332f1oeq3d 6846 . . . . . 6 ((𝜑𝑏𝐶) → (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto𝑏))
3427, 33mpbiri 258 . . . . 5 ((𝜑𝑏𝐶) → ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
35 resiexg 7935 . . . . . . 7 (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
3635elv 3483 . . . . . 6 ( I ↾ 𝑏) ∈ V
37 f1oeq1 6837 . . . . . 6 (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
3836, 37spcev 3606 . . . . 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 3144 . 2 (𝜑 → ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
428, 9, 413jca 1127 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231   I cid 5582  cres 5691  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  WUnicwun 10738  ndxcnx 17227  Basecbs 17245   Full cful 17956   Faith cfth 17957  SetCatcsetc 18129  ExtStrCatcestrc 18177
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-wun 10740  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-func 17909  df-full 17958  df-fth 17959  df-setc 18130  df-estrc 18178
This theorem is referenced by: (None)
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