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Theorem fullsetcestrc 17873
Description: The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u (𝜑𝑈 ∈ WUni)
funcsetcestrc.o (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
funcsetcestrc.e 𝐸 = (ExtStrCat‘𝑈)
Assertion
Ref Expression
fullsetcestrc (𝜑𝐹(𝑆 Full 𝐸)𝐺)
Distinct variable groups:   𝑥,𝐶   𝜑,𝑥   𝑦,𝐶,𝑥   𝜑,𝑦   𝑥,𝐸
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fullsetcestrc
Dummy variables 𝑎 𝑏 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcsetcestrc.s . . 3 𝑆 = (SetCat‘𝑈)
2 funcsetcestrc.c . . 3 𝐶 = (Base‘𝑆)
3 funcsetcestrc.f . . 3 (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
4 funcsetcestrc.u . . 3 (𝜑𝑈 ∈ WUni)
5 funcsetcestrc.o . . 3 (𝜑 → ω ∈ 𝑈)
6 funcsetcestrc.g . . 3 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
7 funcsetcestrc.e . . 3 𝐸 = (ExtStrCat‘𝑈)
81, 2, 3, 4, 5, 6, 7funcsetcestrc 17871 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 17869 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2740 . . . . . . 7 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 2, 3, 4, 5funcsetcestrclem2 17862 . . . . . . . 8 ((𝜑𝑎𝐶) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 714 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5funcsetcestrclem2 17862 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 713 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) ∈ 𝑈)
16 eqid 2740 . . . . . . 7 (Base‘(𝐹𝑎)) = (Base‘(𝐹𝑎))
17 eqid 2740 . . . . . . 7 (Base‘(𝐹𝑏)) = (Base‘(𝐹𝑏))
187, 10, 11, 13, 15, 16, 17elestrchom 17834 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ :(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏))))
191, 2, 3funcsetcestrclem1 17861 . . . . . . . . . . 11 ((𝜑𝑎𝐶) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2019adantrr 714 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2120fveq2d 6773 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22 eqid 2740 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23221strbas 16919 . . . . . . . . . 10 (𝑎𝐶𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2423ad2antrl 725 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2521, 24eqtr4d 2783 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = 𝑎)
261, 2, 3funcsetcestrclem1 17861 . . . . . . . . . . 11 ((𝜑𝑏𝐶) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2726adantrl 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2827fveq2d 6773 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29 eqid 2740 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30291strbas 16919 . . . . . . . . . 10 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130ad2antll 726 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3228, 31eqtr4d 2783 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = 𝑏)
3325, 32feq23d 6592 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) ↔ :𝑎𝑏))
34 simpr 485 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐶𝑏𝐶))
3534ancomd 462 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑏𝐶𝑎𝐶))
36 elmapg 8603 . . . . . . . . . . . . 13 ((𝑏𝐶𝑎𝐶) → ( ∈ (𝑏m 𝑎) ↔ :𝑎𝑏))
3735, 36syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏m 𝑎) ↔ :𝑎𝑏))
3837biimpar 478 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∈ (𝑏m 𝑎))
39 equequ2 2033 . . . . . . . . . . . 12 (𝑘 = → ( = 𝑘 = ))
4039adantl 482 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) ∧ 𝑘 = ) → ( = 𝑘 = ))
41 eqidd 2741 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → = )
4238, 40, 41rspcedvd 3564 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏m 𝑎) = 𝑘)
431, 2, 3, 4, 5, 6funcsetcestrclem6 17867 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏m 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44433expa 1117 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏m 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
4544eqeq2d 2751 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏m 𝑎)) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4645rexbidva 3227 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (∃𝑘 ∈ (𝑏m 𝑎) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏m 𝑎) = 𝑘))
4746adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → (∃𝑘 ∈ (𝑏m 𝑎) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏m 𝑎) = 𝑘))
4842, 47mpbird 256 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏m 𝑎) = ((𝑎𝐺𝑏)‘𝑘))
49 eqid 2740 . . . . . . . . . . . 12 (Hom ‘𝑆) = (Hom ‘𝑆)
501, 4setcbas 17783 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 = (Base‘𝑆))
512, 50eqtr4id 2799 . . . . . . . . . . . . . . . 16 (𝜑𝐶 = 𝑈)
5251eleq2d 2826 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎𝐶𝑎𝑈))
5352biimpcd 248 . . . . . . . . . . . . . 14 (𝑎𝐶 → (𝜑𝑎𝑈))
5453adantr 481 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
5554impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
5651eleq2d 2826 . . . . . . . . . . . . . . 15 (𝜑 → (𝑏𝐶𝑏𝑈))
5756biimpcd 248 . . . . . . . . . . . . . 14 (𝑏𝐶 → (𝜑𝑏𝑈))
5857adantl 482 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
5958impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
601, 10, 49, 55, 59setchom 17785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏m 𝑎))
6160rexeqdv 3348 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏m 𝑎) = ((𝑎𝐺𝑏)‘𝑘)))
6261adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏m 𝑎) = ((𝑎𝐺𝑏)‘𝑘)))
6348, 62mpbird 256 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
6463ex 413 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:𝑎𝑏 → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6533, 64sylbid 239 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6618, 65sylbid 239 . . . . 5 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6766ralrimiv 3109 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
68 dffo3 6973 . . . 4 ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ∧ ∀ ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
699, 67, 68sylanbrc 583 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
7069ralrimivva 3117 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
712, 11, 49isfull2 17617 . 2 (𝐹(𝑆 Full 𝐸)𝐺 ↔ (𝐹(𝑆 Func 𝐸)𝐺 ∧ ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))))
728, 70, 71sylanbrc 583 1 (𝜑𝐹(𝑆 Full 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wcel 2110  wral 3066  wrex 3067  {csn 4567  cop 4573   class class class wbr 5079  cmpt 5162   I cid 5488  cres 5591  wf 6427  ontowfo 6429  cfv 6431  (class class class)co 7269  cmpo 7271  ωcom 7701  m cmap 8590  WUnicwun 10449  ndxcnx 16884  Basecbs 16902  Hom chom 16963   Func cfunc 17559   Full cful 17608  SetCatcsetc 17780  ExtStrCatcestrc 17828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-inf2 9369  ax-cnex 10920  ax-resscn 10921  ax-1cn 10922  ax-icn 10923  ax-addcl 10924  ax-addrcl 10925  ax-mulcl 10926  ax-mulrcl 10927  ax-mulcom 10928  ax-addass 10929  ax-mulass 10930  ax-distr 10931  ax-i2m1 10932  ax-1ne0 10933  ax-1rid 10934  ax-rnegex 10935  ax-rrecex 10936  ax-cnre 10937  ax-pre-lttri 10938  ax-pre-lttrn 10939  ax-pre-ltadd 10940  ax-pre-mulgt0 10941
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-1st 7818  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-1o 8282  df-oadd 8286  df-omul 8287  df-er 8473  df-ec 8475  df-qs 8479  df-map 8592  df-pm 8593  df-ixp 8661  df-en 8709  df-dom 8710  df-sdom 8711  df-fin 8712  df-wun 10451  df-ni 10621  df-pli 10622  df-mi 10623  df-lti 10624  df-plpq 10657  df-mpq 10658  df-ltpq 10659  df-enq 10660  df-nq 10661  df-erq 10662  df-plq 10663  df-mq 10664  df-1nq 10665  df-rq 10666  df-ltnq 10667  df-np 10730  df-plp 10732  df-ltp 10734  df-enr 10804  df-nr 10805  df-c 10870  df-pnf 11004  df-mnf 11005  df-xr 11006  df-ltxr 11007  df-le 11008  df-sub 11199  df-neg 11200  df-nn 11966  df-2 12028  df-3 12029  df-4 12030  df-5 12031  df-6 12032  df-7 12033  df-8 12034  df-9 12035  df-n0 12226  df-z 12312  df-dec 12429  df-uz 12574  df-fz 13231  df-struct 16838  df-slot 16873  df-ndx 16885  df-base 16903  df-hom 16976  df-cco 16977  df-cat 17367  df-cid 17368  df-func 17563  df-full 17610  df-setc 17781  df-estrc 17829
This theorem is referenced by: (None)
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