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Theorem fullsetcestrc 17027
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 17025 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 17023 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2760 . . . . . . 7 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 2, 3, 4, 5funcsetcestrclem2 17016 . . . . . . . 8 ((𝜑𝑎𝐶) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 755 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5funcsetcestrclem2 17016 . . . . . . . 8 ((𝜑𝑏𝐶) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 754 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) ∈ 𝑈)
16 eqid 2760 . . . . . . 7 (Base‘(𝐹𝑎)) = (Base‘(𝐹𝑎))
17 eqid 2760 . . . . . . 7 (Base‘(𝐹𝑏)) = (Base‘(𝐹𝑏))
187, 10, 11, 13, 15, 16, 17elestrchom 16989 . . . . . 6 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ :(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏))))
191, 2, 3funcsetcestrclem1 17015 . . . . . . . . . . 11 ((𝜑𝑎𝐶) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2019adantrr 755 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑎) = {⟨(Base‘ndx), 𝑎⟩})
2120fveq2d 6357 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
22 eqid 2760 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑎⟩} = {⟨(Base‘ndx), 𝑎⟩}
23221strbas 16202 . . . . . . . . . 10 (𝑎𝐶𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2423ad2antrl 766 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎 = (Base‘{⟨(Base‘ndx), 𝑎⟩}))
2521, 24eqtr4d 2797 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑎)) = 𝑎)
261, 2, 3funcsetcestrclem1 17015 . . . . . . . . . . 11 ((𝜑𝑏𝐶) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2726adantrl 754 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝐹𝑏) = {⟨(Base‘ndx), 𝑏⟩})
2827fveq2d 6357 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
29 eqid 2760 . . . . . . . . . . 11 {⟨(Base‘ndx), 𝑏⟩} = {⟨(Base‘ndx), 𝑏⟩}
30291strbas 16202 . . . . . . . . . 10 (𝑏𝐶𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3130ad2antll 767 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏 = (Base‘{⟨(Base‘ndx), 𝑏⟩}))
3228, 31eqtr4d 2797 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (Base‘(𝐹𝑏)) = 𝑏)
3325, 32feq23d 6201 . . . . . . 7 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (:(Base‘(𝐹𝑎))⟶(Base‘(𝐹𝑏)) ↔ :𝑎𝑏))
34 simpr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐶𝑏𝐶))
3534ancomd 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑏𝐶𝑎𝐶))
36 elmapg 8038 . . . . . . . . . . . . 13 ((𝑏𝐶𝑎𝐶) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3735, 36syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏𝑚 𝑎) ↔ :𝑎𝑏))
3837biimpar 503 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∈ (𝑏𝑚 𝑎))
39 equequ2 2108 . . . . . . . . . . . 12 (𝑘 = → ( = 𝑘 = ))
4039adantl 473 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) ∧ 𝑘 = ) → ( = 𝑘 = ))
41 eqidd 2761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → = )
4238, 40, 41rspcedvd 3456 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = 𝑘)
431, 2, 3, 4, 5, 6funcsetcestrclem6 17021 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
44433expa 1112 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
4544eqeq2d 2770 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4645rexbidva 3187 . . . . . . . . . . 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 247 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑏𝑚 𝑎) = ((𝑎𝐺𝑏)‘𝑘))
49 eqid 2760 . . . . . . . . . . . 12 (Hom ‘𝑆) = (Hom ‘𝑆)
501, 4setcbas 16949 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 = (Base‘𝑆))
5150, 2syl6reqr 2813 . . . . . . . . . . . . . . . 16 (𝜑𝐶 = 𝑈)
5251eleq2d 2825 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎𝐶𝑎𝑈))
5352biimpcd 239 . . . . . . . . . . . . . 14 (𝑎𝐶 → (𝜑𝑎𝑈))
5453adantr 472 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
5554impcom 445 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
5651eleq2d 2825 . . . . . . . . . . . . . . 15 (𝜑 → (𝑏𝐶𝑏𝑈))
5756biimpcd 239 . . . . . . . . . . . . . 14 (𝑏𝐶 → (𝜑𝑏𝑈))
5857adantl 473 . . . . . . . . . . . . 13 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
5958impcom 445 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
601, 10, 49, 55, 59setchom 16951 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏𝑚 𝑎))
6160rexeqdv 3284 . . . . . . . . . 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 247 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ :𝑎𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
6463ex 449 . . . . . . 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 3103 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
68 dffo3 6538 . . . 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 701 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
7069ralrimivva 3109 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–onto→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
712, 11, 49isfull2 16792 . 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 701 1 (𝜑𝐹(𝑆 Full 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {csn 4321  cop 4327   class class class wbr 4804  cmpt 4881   I cid 5173  cres 5268  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231  𝑚 cmap 8025  WUnicwun 9734  ndxcnx 16076  Basecbs 16079  Hom chom 16174   Func cfunc 16735   Full cful 16783  SetCatcsetc 16946  ExtStrCatcestrc 16983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-omul 7735  df-er 7913  df-ec 7915  df-qs 7919  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-wun 9736  df-ni 9906  df-pli 9907  df-mi 9908  df-lti 9909  df-plpq 9942  df-mpq 9943  df-ltpq 9944  df-enq 9945  df-nq 9946  df-erq 9947  df-plq 9948  df-mq 9949  df-1nq 9950  df-rq 9951  df-ltnq 9952  df-np 10015  df-plp 10017  df-ltp 10019  df-enr 10089  df-nr 10090  df-c 10154  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-cat 16550  df-cid 16551  df-func 16739  df-full 16785  df-setc 16947  df-estrc 16984
This theorem is referenced by: (None)
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