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Theorem fthsetcestrc 17006
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 faithful. (Contributed by AV, 31-Mar-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
fthsetcestrc (𝜑𝐹(𝑆 Faith 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 fthsetcestrc
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 17005 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 17003 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2760 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 4setcbas 16929 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝑆))
1312, 2syl6reqr 2813 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 = 𝑈)
1413eleq2d 2825 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐶𝑎𝑈))
1514biimpcd 239 . . . . . . . . . . . . . . 15 (𝑎𝐶 → (𝜑𝑎𝑈))
1615adantr 472 . . . . . . . . . . . . . 14 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
1716impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
1813eleq2d 2825 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐶𝑏𝑈))
1918biimpcd 239 . . . . . . . . . . . . . . 15 (𝑏𝐶 → (𝜑𝑏𝑈))
2019adantl 473 . . . . . . . . . . . . . 14 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
2120impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
221, 10, 11, 17, 21setchom 16931 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏𝑚 𝑎))
2322eleq2d 2825 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑎(Hom ‘𝑆)𝑏) ↔ ∈ (𝑏𝑚 𝑎)))
241, 2, 3, 4, 5, 6funcsetcestrclem6 17001 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘) = )
25243expia 1115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏𝑚 𝑎) → ((𝑎𝐺𝑏)‘) = ))
2623, 25sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2726com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ((𝑎𝐺𝑏)‘) = ))
2827adantr 472 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)) → ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ((𝑎𝐺𝑏)‘) = ))
2928impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3022eleq2d 2825 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) ↔ 𝑘 ∈ (𝑏𝑚 𝑎)))
311, 2, 3, 4, 5, 6funcsetcestrclem6 17001 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
32313expia 1115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑘 ∈ (𝑏𝑚 𝑎) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3330, 32sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3433com12 32 . . . . . . . . 9 (𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3534adantl 473 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)) → ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3635impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3729, 36eqeq12d 2775 . . . . . 6 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837biimpd 219 . . . . 5 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
3938ralrimivva 3109 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ (𝑎(Hom ‘𝑆)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
40 dff13 6675 . . . 4 ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)) ∧ ∀ ∈ (𝑎(Hom ‘𝑆)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘)))
419, 39, 40sylanbrc 701 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
4241ralrimivva 3109 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
43 eqid 2760 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
442, 11, 43isfth2 16776 . 2 (𝐹(𝑆 Faith 𝐸)𝐺 ↔ (𝐹(𝑆 Func 𝐸)𝐺 ∧ ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏))))
458, 42, 44sylanbrc 701 1 (𝜑𝐹(𝑆 Faith 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {csn 4321  cop 4327   class class class wbr 4804  cmpt 4881   I cid 5173  cres 5268  wf 6045  1-1wf1 6046  cfv 6049  (class class class)co 6813  cmpt2 6815  ωcom 7230  𝑚 cmap 8023  WUnicwun 9714  ndxcnx 16056  Basecbs 16059  Hom chom 16154   Func cfunc 16715   Faith cfth 16764  SetCatcsetc 16926  ExtStrCatcestrc 16963
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 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
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 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-ec 7913  df-qs 7917  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-wun 9716  df-ni 9886  df-pli 9887  df-mi 9888  df-lti 9889  df-plpq 9922  df-mpq 9923  df-ltpq 9924  df-enq 9925  df-nq 9926  df-erq 9927  df-plq 9928  df-mq 9929  df-1nq 9930  df-rq 9931  df-ltnq 9932  df-np 9995  df-plp 9997  df-ltp 9999  df-enr 10069  df-nr 10070  df-c 10134  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-hom 16168  df-cco 16169  df-cat 16530  df-cid 16531  df-func 16719  df-fth 16766  df-setc 16927  df-estrc 16964
This theorem is referenced by:  embedsetcestrc  17008
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