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Theorem fthsetcestrc 16628
 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 16627 . 2 (𝜑𝐹(𝑆 Func 𝐸)𝐺)
91, 2, 3, 4, 5, 6, 7funcsetcestrclem8 16625 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)⟶((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
104adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑈 ∈ WUni)
11 eqid 2610 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 4setcbas 16551 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝑆))
1312, 2syl6reqr 2663 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 = 𝑈)
1413eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐶𝑎𝑈))
1514biimpcd 238 . . . . . . . . . . . . . . 15 (𝑎𝐶 → (𝜑𝑎𝑈))
1615adantr 480 . . . . . . . . . . . . . 14 ((𝑎𝐶𝑏𝐶) → (𝜑𝑎𝑈))
1716impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑈)
1813eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐶𝑏𝑈))
1918biimpcd 238 . . . . . . . . . . . . . . 15 (𝑏𝐶 → (𝜑𝑏𝑈))
2019adantl 481 . . . . . . . . . . . . . 14 ((𝑎𝐶𝑏𝐶) → (𝜑𝑏𝑈))
2120impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑈)
221, 10, 11, 17, 21setchom 16553 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎(Hom ‘𝑆)𝑏) = (𝑏𝑚 𝑎))
2322eleq2d 2673 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑎(Hom ‘𝑆)𝑏) ↔ ∈ (𝑏𝑚 𝑎)))
241, 2, 3, 4, 5, 6funcsetcestrclem6 16623 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘) = )
25243expia 1259 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑏𝑚 𝑎) → ((𝑎𝐺𝑏)‘) = ))
2623, 25sylbid 229 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ( ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2726com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝑆)𝑏) → ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ((𝑎𝐺𝑏)‘) = ))
2827adantr 480 . . . . . . . 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 2673 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏) ↔ 𝑘 ∈ (𝑏𝑚 𝑎)))
311, 2, 3, 4, 5, 6funcsetcestrclem6 16623 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐶𝑏𝐶) ∧ 𝑘 ∈ (𝑏𝑚 𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
32313expia 1259 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑘 ∈ (𝑏𝑚 𝑎) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3330, 32sylbid 229 . . . . . . . . . 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 481 . . . . . . . 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 2625 . . . . . 6 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837biimpd 218 . . . . 5 (((𝜑 ∧ (𝑎𝐶𝑏𝐶)) ∧ ( ∈ (𝑎(Hom ‘𝑆)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
3938ralrimivva 2954 . . . 4 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → ∀ ∈ (𝑎(Hom ‘𝑆)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝑆)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
40 dff13 6416 . . . 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 695 . . 3 ((𝜑 ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
4241ralrimivva 2954 . 2 (𝜑 → ∀𝑎𝐶𝑏𝐶 (𝑎𝐺𝑏):(𝑎(Hom ‘𝑆)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝐸)(𝐹𝑏)))
43 eqid 2610 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
442, 11, 43isfth2 16398 . 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 695 1 (𝜑𝐹(𝑆 Faith 𝐸)𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ↑𝑚 cmap 7744  WUnicwun 9401  ndxcnx 15692  Basecbs 15695  Hom chom 15779   Func cfunc 16337   Faith cfth 16386  SetCatcsetc 16548  ExtStrCatcestrc 16585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-c 9821  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-fth 16388  df-setc 16549  df-estrc 16586 This theorem is referenced by:  embedsetcestrc  16630
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