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Theorem fthestrcsetc 16837
Description: The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is faithful. (Contributed by AV, 2-Apr-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b 𝐵 = (Base‘𝐸)
funcestrcsetc.c 𝐶 = (Base‘𝑆)
funcestrcsetc.u (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
Assertion
Ref Expression
fthestrcsetc (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fthestrcsetc
Dummy variables 𝑎 𝑏 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcestrcsetc.e . . 3 𝐸 = (ExtStrCat‘𝑈)
2 funcestrcsetc.s . . 3 𝑆 = (SetCat‘𝑈)
3 funcestrcsetc.b . . 3 𝐵 = (Base‘𝐸)
4 funcestrcsetc.c . . 3 𝐶 = (Base‘𝑆)
5 funcestrcsetc.u . . 3 (𝜑𝑈 ∈ WUni)
6 funcestrcsetc.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcestrcsetc.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
81, 2, 3, 4, 5, 6, 7funcestrcsetc 16836 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 16834 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2651 . . . . . . . . . . . . 13 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 5estrcbas 16812 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐸))
1312, 3syl6reqr 2704 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = 𝑈)
1413eleq2d 2716 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐵𝑎𝑈))
1514biimpcd 239 . . . . . . . . . . . . . . 15 (𝑎𝐵 → (𝜑𝑎𝑈))
1615adantr 480 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
1716impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
1813eleq2d 2716 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐵𝑏𝑈))
1918biimpcd 239 . . . . . . . . . . . . . . 15 (𝑏𝐵 → (𝜑𝑏𝑈))
2019adantl 481 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
2120impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
22 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝑎) = (Base‘𝑎)
23 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝑏) = (Base‘𝑏)
241, 10, 11, 17, 21, 22, 23estrchom 16814 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2524eleq2d 2716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16832 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘) = )
27263expia 1286 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘) = ))
2825, 27sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2928com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3029adantr 480 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3130impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3224eleq2d 2716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16832 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34333expia 1286 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3532, 34sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3635com12 32 . . . . . . . . 9 (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3736adantl 481 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3837impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3931, 38eqeq12d 2666 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4039biimpd 219 . . . . 5 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
4140ralrimivva 3000 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
42 dff13 6552 . . . 4 ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ∧ ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘)))
439, 41, 42sylanbrc 699 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
4443ralrimivva 3000 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
45 eqid 2651 . . 3 (Hom ‘𝑆) = (Hom ‘𝑆)
463, 11, 45isfth2 16622 . 2 (𝐹(𝐸 Faith 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))))
478, 44, 46sylanbrc 699 1 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  cmpt 4762   I cid 5052  cres 5145  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑚 cmap 7899  WUnicwun 9560  Basecbs 15904  Hom chom 15999   Func cfunc 16561   Faith cfth 16610  SetCatcsetc 16772  ExtStrCatcestrc 16809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-wun 9562  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-func 16565  df-fth 16612  df-setc 16773  df-estrc 16810
This theorem is referenced by:  equivestrcsetc  16839
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