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Theorem fullsetcestrc 16727
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 16725 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 16723 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2621 . . . . . . 7 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 2, 3, 4, 5funcsetcestrclem2 16716 . . . . . . . 8 ((𝜑𝑎𝐶) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 752 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5funcsetcestrclem2 16716 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 751 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) ∈ 𝑈)
16 eqid 2621 . . . . . . 7 (Base‘(𝐹𝑎)) = (Base‘(𝐹𝑎))
17 eqid 2621 . . . . . . 7 (Base‘(𝐹𝑏)) = (Base‘(𝐹𝑏))
187, 10, 11, 13, 15, 16, 17elestrchom 16689 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ :(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏))))
191, 2, 3funcsetcestrclem1 16715 . . . . . . . . . . 11 ((𝜑𝑎𝐶) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2019adantrr 752 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2120fveq2d 6152 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22 eqid 2621 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23221strbas 15901 . . . . . . . . . 10 (𝑎𝐶𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2423ad2antrl 763 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2521, 24eqtr4d 2658 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = 𝑎)
261, 2, 3funcsetcestrclem1 16715 . . . . . . . . . . 11 ((𝜑𝑏𝐶) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2726adantrl 751 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2827fveq2d 6152 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29 eqid 2621 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30291strbas 15901 . . . . . . . . . 10 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130ad2antll 764 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3228, 31eqtr4d 2658 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = 𝑏)
3325, 32feq23d 5997 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) ↔ :𝑎𝑏))
34 simpr 477 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐶𝑏𝐶))
3534ancomd 467 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑏𝐶𝑎𝐶))
36 elmapg 7815 . . . . . . . . . . . . 13 ((𝑏𝐶𝑎𝐶) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3735, 36syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3837biimpar 502 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∈ (𝑏𝑚 𝑎))
39 equequ2 1950 . . . . . . . . . . . 12 (𝑘 = → ( = 𝑘 = ))
4039adantl 482 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) ∧ 𝑘 = ) → ( = 𝑘 = ))
41 eqidd 2622 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → = )
4238, 40, 41rspcedvd 3302 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = 𝑘)
431, 2, 3, 4, 5, 6funcsetcestrclem6 16721 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44433expa 1262 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
4544eqeq2d 2631 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4645rexbidva 3042 . . . . . . . . . . 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 247 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘))
49 eqid 2621 . . . . . . . . . . . 12 (Hom ‘𝑆) = (Hom ‘𝑆)
501, 4setcbas 16649 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 = (Base‘𝑆))
5150, 2syl6reqr 2674 . . . . . . . . . . . . . . . 16 (𝜑𝐶 = 𝑈)
5251eleq2d 2684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎𝐶𝑎𝑈))
5352biimpcd 239 . . . . . . . . . . . . . 14 (𝑎𝐶 → (𝜑𝑎𝑈))
5453adantr 481 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
5554impcom 446 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
5651eleq2d 2684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑏𝐶𝑏𝑈))
5756biimpcd 239 . . . . . . . . . . . . . 14 (𝑏𝐶 → (𝜑𝑏𝑈))
5857adantl 482 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
5958impcom 446 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
601, 10, 49, 55, 59setchom 16651 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏𝑚 𝑎))
6160rexeqdv 3134 . . . . . . . . . 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 247 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
6463ex 450 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:𝑎𝑏 → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6533, 64sylbid 230 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6618, 65sylbid 230 . . . . 5 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
6766ralrimiv 2959 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
68 dffo3 6330 . . . 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 697 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
7069ralrimivva 2965 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
712, 11, 49isfull2 16492 . 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 697 1 (𝜑𝐹(𝑆 Full 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {csn 4148  cop 4154   class class class wbr 4613  cmpt 4673   I cid 4984  cres 5076  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  𝑚 cmap 7802  WUnicwun 9466  ndxcnx 15778  Basecbs 15781  Hom chom 15873   Func cfunc 16435   Full cful 16483  SetCatcsetc 16646  ExtStrCatcestrc 16683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wun 9468  df-ni 9638  df-pli 9639  df-mi 9640  df-lti 9641  df-plpq 9674  df-mpq 9675  df-ltpq 9676  df-enq 9677  df-nq 9678  df-erq 9679  df-plq 9680  df-mq 9681  df-1nq 9682  df-rq 9683  df-ltnq 9684  df-np 9747  df-plp 9749  df-ltp 9751  df-enr 9821  df-nr 9822  df-c 9886  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-func 16439  df-full 16485  df-setc 16647  df-estrc 16684
This theorem is referenced by: (None)
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