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Theorem equivestrcsetc 16556
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 16554 . 2 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7fullestrcsetc 16555 . 2 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
102, 5setcbas 16492 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝑆))
1110, 4syl6reqr 2657 . . . . . . . 8 (𝜑𝐶 = 𝑈)
1211eleq2d 2667 . . . . . . 7 (𝜑 → (𝑏𝐶𝑏𝑈))
13 eqid 2604 . . . . . . . . 9 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
14 equivestrcsetc.i . . . . . . . . 9 (𝜑 → (Base‘ndx) ∈ 𝑈)
1513, 5, 141strwunbndx 15748 . . . . . . . 8 ((𝜑𝑏𝑈) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
1615ex 448 . . . . . . 7 (𝜑 → (𝑏𝑈 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1712, 16sylbid 228 . . . . . 6 (𝜑 → (𝑏𝐶 → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
1817imp 443 . . . . 5 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈)
191, 5estrcbas 16529 . . . . . . . 8 (𝜑𝑈 = (Base‘𝐸))
2019adantr 479 . . . . . . 7 ((𝜑𝑏𝐶) → 𝑈 = (Base‘𝐸))
2120, 3syl6reqr 2657 . . . . . 6 ((𝜑𝑏𝐶) → 𝐵 = 𝑈)
2221eleq2d 2667 . . . . 5 ((𝜑𝑏𝐶) → ({⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵 ↔ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝑈))
2318, 22mpbird 245 . . . 4 ((𝜑𝑏𝐶) → {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵)
24 fveq2 6083 . . . . . . 7 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝐹𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
25 f1oeq3 6022 . . . . . . 7 ((𝐹𝑎) = (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) → (𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2624, 25syl 17 . . . . . 6 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2726exbidv 1835 . . . . 5 (𝑎 = {⟨(Base‘ndx), 𝑏⟩} → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
2827adantl 480 . . . 4 (((𝜑𝑏𝐶) ∧ 𝑎 = {⟨(Base‘ndx), 𝑏⟩}) → (∃𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎) ↔ ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
29 f1oi 6066 . . . . . 6 ( I ↾ 𝑏):𝑏1-1-onto𝑏
301, 2, 3, 4, 5, 6funcestrcsetclem1 16544 . . . . . . . . 9 ((𝜑 ∧ {⟨(Base‘ndx), 𝑏⟩} ∈ 𝐵) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3123, 30syldan 485 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
32131strbas 15747 . . . . . . . . 9 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3332adantl 480 . . . . . . . 8 ((𝜑𝑏𝐶) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3431, 33eqtr4d 2641 . . . . . . 7 ((𝜑𝑏𝐶) → (𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏)
35 f1oeq3 6022 . . . . . . 7 ((𝐹‘{⟨(Base‘ndx), 𝑏⟩}) = 𝑏 → (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto𝑏))
3634, 35syl 17 . . . . . 6 ((𝜑𝑏𝐶) → (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto𝑏))
3729, 36mpbiri 246 . . . . 5 ((𝜑𝑏𝐶) → ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
38 vex 3170 . . . . . . 7 𝑏 ∈ V
39 resiexg 6966 . . . . . . 7 (𝑏 ∈ V → ( I ↾ 𝑏) ∈ V)
4038, 39ax-mp 5 . . . . . 6 ( I ↾ 𝑏) ∈ V
41 f1oeq1 6020 . . . . . 6 (𝑖 = ( I ↾ 𝑏) → (𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) ↔ ( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩})))
4240, 41spcev 3267 . . . . 5 (( I ↾ 𝑏):𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
4337, 42syl 17 . . . 4 ((𝜑𝑏𝐶) → ∃𝑖 𝑖:𝑏1-1-onto→(𝐹‘{⟨(Base‘ndx), 𝑏⟩}))
4423, 28, 43rspcedvd 3283 . . 3 ((𝜑𝑏𝐶) → ∃𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
4544ralrimiva 2943 . 2 (𝜑 → ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎))
468, 9, 453jca 1234 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 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wral 2890  wrex 2891  Vcvv 3167  {csn 4119  cop 4125   class class class wbr 4572  cmpt 4632   I cid 4933  cres 5025  1-1-ontowf1o 5784  cfv 5785  (class class class)co 6522  cmpt2 6524  𝑚 cmap 7716  WUnicwun 9373  ndxcnx 15633  Basecbs 15636   Full cful 16326   Faith cfth 16327  SetCatcsetc 16489  ExtStrCatcestrc 16526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-ixp 7767  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-wun 9375  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-3 10922  df-4 10923  df-5 10924  df-6 10925  df-7 10926  df-8 10927  df-9 10928  df-n0 11135  df-z 11206  df-dec 11321  df-uz 11515  df-fz 12148  df-struct 15638  df-ndx 15639  df-slot 15640  df-base 15641  df-hom 15734  df-cco 15735  df-cat 16093  df-cid 16094  df-func 16282  df-full 16328  df-fth 16329  df-setc 16490  df-estrc 16527
This theorem is referenced by: (None)
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