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Theorem fullestrcsetc 18196
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 18194 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 18192 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2737 . . . . . . . 8 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 2, 3, 4, 5, 6funcestrcsetclem2 18186 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 717 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5, 6funcestrcsetclem2 18186 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 716 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) ∈ 𝑈)
162, 10, 11, 13, 15elsetchom 18126 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(𝐹𝑎)⟶(𝐹𝑏)))
171, 2, 3, 4, 5, 6funcestrcsetclem1 18185 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) = (Base‘𝑎))
1817adantrr 717 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) = (Base‘𝑎))
191, 2, 3, 4, 5, 6funcestrcsetclem1 18185 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) = (Base‘𝑏))
2019adantrl 716 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) = (Base‘𝑏))
2118, 20feq23d 6731 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(𝐹𝑎)⟶(𝐹𝑏) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2216, 21bitrd 279 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
23 fvex 6919 . . . . . . . . . . . . 13 (Base‘𝑏) ∈ V
24 fvex 6919 . . . . . . . . . . . . 13 (Base‘𝑎) ∈ V
2523, 24pm3.2i 470 . . . . . . . . . . . 12 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26 elmapg 8879 . . . . . . . . . . . 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 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
29 equequ2 2025 . . . . . . . . . . 11 (𝑘 = → ( = 𝑘 = ))
3029adantl 481 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ) → ( = 𝑘 = ))
31 eqidd 2738 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → = )
3228, 30, 31rspcedvd 3624 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘)
33 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝑎) = (Base‘𝑎)
34 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝑏) = (Base‘𝑏)
351, 2, 3, 4, 5, 6, 7, 33, 34funcestrcsetclem6 18190 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36353expa 1119 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3736eqeq2d 2748 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837rexbidva 3177 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘))
3938adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = 𝑘))
4032, 39mpbird 257 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘))
41 eqid 2737 . . . . . . . . . . 11 (Hom ‘𝐸) = (Hom ‘𝐸)
421, 5estrcbas 18169 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐸))
433, 42eqtr4id 2796 . . . . . . . . . . . . . . 15 (𝜑𝐵 = 𝑈)
4443eleq2d 2827 . . . . . . . . . . . . . 14 (𝜑 → (𝑎𝐵𝑎𝑈))
4544biimpcd 249 . . . . . . . . . . . . 13 (𝑎𝐵 → (𝜑𝑎𝑈))
4645adantr 480 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
4746impcom 407 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
4843eleq2d 2827 . . . . . . . . . . . . . 14 (𝜑 → (𝑏𝐵𝑏𝑈))
4948biimpcd 249 . . . . . . . . . . . . 13 (𝑏𝐵 → (𝜑𝑏𝑈))
5049adantl 481 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
5150impcom 407 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
521, 10, 41, 47, 51, 33, 34estrchom 18171 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
5352rexeqdv 3327 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5453adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5540, 54mpbird 257 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
5655ex 412 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5722, 56sylbid 240 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5857ralrimiv 3145 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
59 dffo3 7122 . . . 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 583 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
6160ralrimivva 3202 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
623, 11, 41isfull2 17958 . 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 583 1 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225   I cid 5577  cres 5687  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  WUnicwun 10740  Basecbs 17247  Hom chom 17308   Func cfunc 17899   Full cful 17949  SetCatcsetc 18120  ExtStrCatcestrc 18166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-wun 10742  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-func 17903  df-full 17951  df-setc 18121  df-estrc 18167
This theorem is referenced by:  equivestrcsetc  18197
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