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Theorem equivestrcsetc 18045
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 = (ExtStrCatβ€˜π‘ˆ)
funcestrcsetc.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcestrcsetc.b 𝐡 = (Baseβ€˜πΈ)
funcestrcsetc.c 𝐢 = (Baseβ€˜π‘†)
funcestrcsetc.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcestrcsetc.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcestrcsetc.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
equivestrcsetc.i (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
Assertion
Ref Expression
equivestrcsetc (πœ‘ β†’ (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ βˆ€π‘ ∈ 𝐢 βˆƒπ‘Ž ∈ 𝐡 βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž)))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   πœ‘,𝑦   π‘Ž,𝑏,π‘₯,𝑦,𝐡   𝐹,π‘Ž,𝑏   𝐺,π‘Ž,𝑏   𝐸,π‘Ž,𝑏   𝑆,π‘Ž,𝑏   πœ‘,π‘Ž,𝑏   𝐢,π‘Ž   𝑖,𝐹,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(𝑖)   𝐡(𝑖)   𝐢(𝑦,𝑖,𝑏)   𝑆(π‘₯,𝑦,𝑖)   π‘ˆ(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝐸(π‘₯,𝑦,𝑖)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦,𝑖)

Proof of Theorem equivestrcsetc
StepHypRef Expression
1 funcestrcsetc.e . . 3 𝐸 = (ExtStrCatβ€˜π‘ˆ)
2 funcestrcsetc.s . . 3 𝑆 = (SetCatβ€˜π‘ˆ)
3 funcestrcsetc.b . . 3 𝐡 = (Baseβ€˜πΈ)
4 funcestrcsetc.c . . 3 𝐢 = (Baseβ€˜π‘†)
5 funcestrcsetc.u . . 3 (πœ‘ β†’ π‘ˆ ∈ WUni)
6 funcestrcsetc.f . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
7 funcestrcsetc.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
81, 2, 3, 4, 5, 6, 7fthestrcsetc 18043 . 2 (πœ‘ β†’ 𝐹(𝐸 Faith 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7fullestrcsetc 18044 . 2 (πœ‘ β†’ 𝐹(𝐸 Full 𝑆)𝐺)
102, 5setcbas 17969 . . . . . . . . 9 (πœ‘ β†’ π‘ˆ = (Baseβ€˜π‘†))
114, 10eqtr4id 2792 . . . . . . . 8 (πœ‘ β†’ 𝐢 = π‘ˆ)
1211eleq2d 2820 . . . . . . 7 (πœ‘ β†’ (𝑏 ∈ 𝐢 ↔ 𝑏 ∈ π‘ˆ))
13 eqid 2733 . . . . . . . . 9 {⟨(Baseβ€˜ndx), π‘βŸ©} = {⟨(Baseβ€˜ndx), π‘βŸ©}
14 equivestrcsetc.i . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
1513, 5, 141strwunbndx 17107 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ π‘ˆ) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ π‘ˆ)
1615ex 414 . . . . . . 7 (πœ‘ β†’ (𝑏 ∈ π‘ˆ β†’ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ π‘ˆ))
1712, 16sylbid 239 . . . . . 6 (πœ‘ β†’ (𝑏 ∈ 𝐢 β†’ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ π‘ˆ))
1817imp 408 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ π‘ˆ)
191, 5estrcbas 18017 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΈ))
2019adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ π‘ˆ = (Baseβ€˜πΈ))
213, 20eqtr4id 2792 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ 𝐡 = π‘ˆ)
2218, 21eleqtrrd 2837 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ 𝐡)
23 fveq2 6843 . . . . . . 7 (π‘Ž = {⟨(Baseβ€˜ndx), π‘βŸ©} β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
2423f1oeq3d 6782 . . . . . 6 (π‘Ž = {⟨(Baseβ€˜ndx), π‘βŸ©} β†’ (𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž) ↔ 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©})))
2524exbidv 1925 . . . . 5 (π‘Ž = {⟨(Baseβ€˜ndx), π‘βŸ©} β†’ (βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž) ↔ βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©})))
2625adantl 483 . . . 4 (((πœ‘ ∧ 𝑏 ∈ 𝐢) ∧ π‘Ž = {⟨(Baseβ€˜ndx), π‘βŸ©}) β†’ (βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž) ↔ βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©})))
27 f1oi 6823 . . . . . 6 ( I β†Ύ 𝑏):𝑏–1-1-onto→𝑏
281, 2, 3, 4, 5, 6funcestrcsetclem1 18033 . . . . . . . . 9 ((πœ‘ ∧ {⟨(Baseβ€˜ndx), π‘βŸ©} ∈ 𝐡) β†’ (πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) = (Baseβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
2922, 28syldan 592 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ (πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) = (Baseβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
30131strbas 17105 . . . . . . . . 9 (𝑏 ∈ 𝐢 β†’ 𝑏 = (Baseβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
3130adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ 𝑏 = (Baseβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
3229, 31eqtr4d 2776 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ (πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) = 𝑏)
3332f1oeq3d 6782 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ (( I β†Ύ 𝑏):𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) ↔ ( I β†Ύ 𝑏):𝑏–1-1-onto→𝑏))
3427, 33mpbiri 258 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ ( I β†Ύ 𝑏):𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
35 resiexg 7852 . . . . . . 7 (𝑏 ∈ V β†’ ( I β†Ύ 𝑏) ∈ V)
3635elv 3450 . . . . . 6 ( I β†Ύ 𝑏) ∈ V
37 f1oeq1 6773 . . . . . 6 (𝑖 = ( I β†Ύ 𝑏) β†’ (𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) ↔ ( I β†Ύ 𝑏):𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©})))
3836, 37spcev 3564 . . . . 5 (( I β†Ύ 𝑏):𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}) β†’ βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
3934, 38syl 17 . . . 4 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜{⟨(Baseβ€˜ndx), π‘βŸ©}))
4022, 26, 39rspcedvd 3582 . . 3 ((πœ‘ ∧ 𝑏 ∈ 𝐢) β†’ βˆƒπ‘Ž ∈ 𝐡 βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž))
4140ralrimiva 3140 . 2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐢 βˆƒπ‘Ž ∈ 𝐡 βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž))
428, 9, 413jca 1129 1 (πœ‘ β†’ (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ βˆ€π‘ ∈ 𝐢 βˆƒπ‘Ž ∈ 𝐡 βˆƒπ‘– 𝑖:𝑏–1-1-ontoβ†’(πΉβ€˜π‘Ž)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444  {csn 4587  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189   I cid 5531   β†Ύ cres 5636  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360   ↑m cmap 8768  WUnicwun 10641  ndxcnx 17070  Basecbs 17088   Full cful 17794   Faith cfth 17795  SetCatcsetc 17966  ExtStrCatcestrc 18014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-wun 10643  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-full 17796  df-fth 17797  df-setc 17967  df-estrc 18015
This theorem is referenced by: (None)
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