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

Theorem fullestrcsetc 17400
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 full. (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
fullestrcsetc (𝜑𝐹(𝐸 Full 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 fullestrcsetc
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 17398 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 17396 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2821 . . . . . . . 8 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 2, 3, 4, 5, 6funcestrcsetclem2 17390 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5, 6funcestrcsetclem2 17390 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 714 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) ∈ 𝑈)
162, 10, 11, 13, 15elsetchom 17340 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(𝐹𝑎)⟶(𝐹𝑏)))
171, 2, 3, 4, 5, 6funcestrcsetclem1 17389 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) = (Base‘𝑎))
1817adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) = (Base‘𝑎))
191, 2, 3, 4, 5, 6funcestrcsetclem1 17389 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) = (Base‘𝑏))
2019adantrl 714 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) = (Base‘𝑏))
2118, 20feq23d 6508 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(𝐹𝑎)⟶(𝐹𝑏) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2216, 21bitrd 281 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
23 fvex 6682 . . . . . . . . . . . . 13 (Base‘𝑏) ∈ V
24 fvex 6682 . . . . . . . . . . . . 13 (Base‘𝑎) ∈ V
2523, 24pm3.2i 473 . . . . . . . . . . . 12 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26 elmapg 8418 . . . . . . . . . . . 12 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2725, 26mp1i 13 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2827biimpar 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
29 equequ2 2029 . . . . . . . . . . 11 (𝑘 = → ( = 𝑘 = ))
3029adantl 484 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ) → ( = 𝑘 = ))
31 eqidd 2822 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → = )
3228, 30, 31rspcedvd 3625 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘)
33 eqid 2821 . . . . . . . . . . . . . 14 (Base‘𝑎) = (Base‘𝑎)
34 eqid 2821 . . . . . . . . . . . . . 14 (Base‘𝑏) = (Base‘𝑏)
351, 2, 3, 4, 5, 6, 7, 33, 34funcestrcsetclem6 17394 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36353expa 1114 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3736eqeq2d 2832 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837rexbidva 3296 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘))
3938adantr 483 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘))
4032, 39mpbird 259 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘))
41 eqid 2821 . . . . . . . . . . 11 (Hom ‘𝐸) = (Hom ‘𝐸)
421, 5estrcbas 17374 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐸))
4342, 3syl6reqr 2875 . . . . . . . . . . . . . . 15 (𝜑𝐵 = 𝑈)
4443eleq2d 2898 . . . . . . . . . . . . . 14 (𝜑 → (𝑎𝐵𝑎𝑈))
4544biimpcd 251 . . . . . . . . . . . . 13 (𝑎𝐵 → (𝜑𝑎𝑈))
4645adantr 483 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
4746impcom 410 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
4843eleq2d 2898 . . . . . . . . . . . . . 14 (𝜑 → (𝑏𝐵𝑏𝑈))
4948biimpcd 251 . . . . . . . . . . . . 13 (𝑏𝐵 → (𝜑𝑏𝑈))
5049adantl 484 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
5150impcom 410 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
521, 10, 41, 47, 51, 33, 34estrchom 17376 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
5352rexeqdv 3416 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5453adantr 483 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5540, 54mpbird 259 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
5655ex 415 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5722, 56sylbid 242 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5857ralrimiv 3181 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
59 dffo3 6867 . . . 4 ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ ((𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ∧ ∀ ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
609, 58, 59sylanbrc 585 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
6160ralrimivva 3191 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
623, 11, 41isfull2 17180 . 2 (𝐹(𝐸 Full 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))))
638, 61, 62sylanbrc 585 1 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3494   class class class wbr 5065  cmpt 5145   I cid 5458  cres 5556  wf 6350  ontowfo 6352  cfv 6354  (class class class)co 7155  cmpo 7157  m cmap 8405  WUnicwun 10121  Basecbs 16482  Hom chom 16575   Func cfunc 17123   Full cful 17171  SetCatcsetc 17334  ExtStrCatcestrc 17371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-wun 10123  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-fz 12892  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-hom 16588  df-cco 16589  df-cat 16938  df-cid 16939  df-func 17127  df-full 17173  df-setc 17335  df-estrc 17372
This theorem is referenced by:  equivestrcsetc  17401
  Copyright terms: Public domain W3C validator