MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fthestrcsetc Structured version   Visualization version   GIF version

Theorem fthestrcsetc 17105
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 17104 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 17102 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 473 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2799 . . . . . . . . . . . . 13 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 5estrcbas 17079 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐸))
1312, 3syl6reqr 2852 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = 𝑈)
1413eleq2d 2864 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐵𝑎𝑈))
1514biimpcd 241 . . . . . . . . . . . . . . 15 (𝑎𝐵 → (𝜑𝑎𝑈))
1615adantr 473 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
1716impcom 397 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
1813eleq2d 2864 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐵𝑏𝑈))
1918biimpcd 241 . . . . . . . . . . . . . . 15 (𝑏𝐵 → (𝜑𝑏𝑈))
2019adantl 474 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
2120impcom 397 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
22 eqid 2799 . . . . . . . . . . . . 13 (Base‘𝑎) = (Base‘𝑎)
23 eqid 2799 . . . . . . . . . . . . 13 (Base‘𝑏) = (Base‘𝑏)
241, 10, 11, 17, 21, 22, 23estrchom 17081 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2524eleq2d 2864 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 17100 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘) = )
27263expia 1151 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘) = ))
2825, 27sylbid 232 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2928com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3029adantr 473 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3130impcom 397 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3224eleq2d 2864 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 17100 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34333expia 1151 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3532, 34sylbid 232 . . . . . . . . . 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 474 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3837impcom 397 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3931, 38eqeq12d 2814 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4039biimpd 221 . . . . 5 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
4140ralrimivva 3152 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
42 dff13 6740 . . . 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 579 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
4443ralrimivva 3152 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
45 eqid 2799 . . 3 (Hom ‘𝑆) = (Hom ‘𝑆)
463, 11, 45isfth2 16889 . 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 579 1 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089   class class class wbr 4843  cmpt 4922   I cid 5219  cres 5314  wf 6097  1-1wf1 6098  cfv 6101  (class class class)co 6878  cmpt2 6880  𝑚 cmap 8095  WUnicwun 9810  Basecbs 16184  Hom chom 16278   Func cfunc 16828   Faith cfth 16877  SetCatcsetc 17039  ExtStrCatcestrc 17076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-ixp 8149  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-wun 9812  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-dec 11784  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-hom 16291  df-cco 16292  df-cat 16643  df-cid 16644  df-func 16832  df-fth 16879  df-setc 17040  df-estrc 17077
This theorem is referenced by:  equivestrcsetc  17107
  Copyright terms: Public domain W3C validator