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Theorem fullsetcestrc 17086
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 17084 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 17082 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2765 . . . . . . 7 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 2, 3, 4, 5funcsetcestrclem2 17075 . . . . . . . 8 ((𝜑𝑎𝐶) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 708 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5funcsetcestrclem2 17075 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 707 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) ∈ 𝑈)
16 eqid 2765 . . . . . . 7 (Base‘(𝐹𝑎)) = (Base‘(𝐹𝑎))
17 eqid 2765 . . . . . . 7 (Base‘(𝐹𝑏)) = (Base‘(𝐹𝑏))
187, 10, 11, 13, 15, 16, 17elestrchom 17047 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ :(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏))))
191, 2, 3funcsetcestrclem1 17074 . . . . . . . . . . 11 ((𝜑𝑎𝐶) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2019adantrr 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2120fveq2d 6383 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22 eqid 2765 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23221strbas 16266 . . . . . . . . . 10 (𝑎𝐶𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2423ad2antrl 719 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2521, 24eqtr4d 2802 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = 𝑎)
261, 2, 3funcsetcestrclem1 17074 . . . . . . . . . . 11 ((𝜑𝑏𝐶) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2726adantrl 707 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2827fveq2d 6383 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29 eqid 2765 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30291strbas 16266 . . . . . . . . . 10 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130ad2antll 720 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3228, 31eqtr4d 2802 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = 𝑏)
3325, 32feq23d 6220 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) ↔ :𝑎𝑏))
34 simpr 477 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐶𝑏𝐶))
3534ancomd 453 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑏𝐶𝑎𝐶))
36 elmapg 8077 . . . . . . . . . . . . 13 ((𝑏𝐶𝑎𝐶) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3735, 36syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3837biimpar 469 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∈ (𝑏𝑚 𝑎))
39 equequ2 2123 . . . . . . . . . . . 12 (𝑘 = → ( = 𝑘 = ))
4039adantl 473 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) ∧ 𝑘 = ) → ( = 𝑘 = ))
41 eqidd 2766 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → = )
4238, 40, 41rspcedvd 3469 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = 𝑘)
431, 2, 3, 4, 5, 6funcsetcestrclem6 17080 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44433expa 1147 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
4544eqeq2d 2775 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4645rexbidva 3196 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏𝑚 𝑎) = 𝑘))
4746adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → (∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏𝑚 𝑎) = 𝑘))
4842, 47mpbird 248 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘))
49 eqid 2765 . . . . . . . . . . . 12 (Hom ‘𝑆) = (Hom ‘𝑆)
501, 4setcbas 17007 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 = (Base‘𝑆))
5150, 2syl6reqr 2818 . . . . . . . . . . . . . . . 16 (𝜑𝐶 = 𝑈)
5251eleq2d 2830 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎𝐶𝑎𝑈))
5352biimpcd 240 . . . . . . . . . . . . . 14 (𝑎𝐶 → (𝜑𝑎𝑈))
5453adantr 472 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
5554impcom 396 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
5651eleq2d 2830 . . . . . . . . . . . . . . 15 (𝜑 → (𝑏𝐶𝑏𝑈))
5756biimpcd 240 . . . . . . . . . . . . . 14 (𝑏𝐶 → (𝜑𝑏𝑈))
5857adantl 473 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
5958impcom 396 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
601, 10, 49, 55, 59setchom 17009 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏𝑚 𝑎))
6160rexeqdv 3293 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘)))
6261adantr 472 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → (∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘)))
6348, 62mpbird 248 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
6463ex 401 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:𝑎𝑏 → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6533, 64sylbid 231 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6618, 65sylbid 231 . . . . 5 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6766ralrimiv 3112 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
68 dffo3 6568 . . . 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 578 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
7069ralrimivva 3118 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
712, 11, 49isfull2 16850 . 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 578 1 (𝜑𝐹(𝑆 Full 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {csn 4336  cop 4342   class class class wbr 4811  cmpt 4890   I cid 5186  cres 5281  wf 6066  ontowfo 6068  cfv 6070  (class class class)co 6846  cmpt2 6848  ωcom 7267  𝑚 cmap 8064  WUnicwun 9779  ndxcnx 16141  Basecbs 16144  Hom chom 16239   Func cfunc 16793   Full cful 16841  SetCatcsetc 17004  ExtStrCatcestrc 17041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-omul 7773  df-er 7951  df-ec 7953  df-qs 7957  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-wun 9781  df-ni 9951  df-pli 9952  df-mi 9953  df-lti 9954  df-plpq 9987  df-mpq 9988  df-ltpq 9989  df-enq 9990  df-nq 9991  df-erq 9992  df-plq 9993  df-mq 9994  df-1nq 9995  df-rq 9996  df-ltnq 9997  df-np 10060  df-plp 10062  df-ltp 10064  df-enr 10134  df-nr 10135  df-c 10199  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-z 11629  df-dec 11746  df-uz 11892  df-fz 12539  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-hom 16252  df-cco 16253  df-cat 16608  df-cid 16609  df-func 16797  df-full 16843  df-setc 17005  df-estrc 17042
This theorem is referenced by: (None)
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