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Theorem fullestrcsetc 17144
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 17142 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 17140 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2825 . . . . . . . 8 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 2, 3, 4, 5, 6funcestrcsetclem2 17134 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 708 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5, 6funcestrcsetclem2 17134 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 707 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) ∈ 𝑈)
162, 10, 11, 13, 15elsetchom 17083 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(𝐹𝑎)⟶(𝐹𝑏)))
171, 2, 3, 4, 5, 6funcestrcsetclem1 17133 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) = (Base‘𝑎))
1817adantrr 708 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) = (Base‘𝑎))
191, 2, 3, 4, 5, 6funcestrcsetclem1 17133 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) = (Base‘𝑏))
2019adantrl 707 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) = (Base‘𝑏))
2118, 20feq23d 6273 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(𝐹𝑎)⟶(𝐹𝑏) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2216, 21bitrd 271 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
23 fvex 6446 . . . . . . . . . . . . 13 (Base‘𝑏) ∈ V
24 fvex 6446 . . . . . . . . . . . . 13 (Base‘𝑎) ∈ V
2523, 24pm3.2i 464 . . . . . . . . . . . 12 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26 elmapg 8135 . . . . . . . . . . . 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 471 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
29 equequ2 2130 . . . . . . . . . . 11 (𝑘 = → ( = 𝑘 = ))
3029adantl 475 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ) → ( = 𝑘 = ))
31 eqidd 2826 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → = )
3228, 30, 31rspcedvd 3533 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘)
33 eqid 2825 . . . . . . . . . . . . . 14 (Base‘𝑎) = (Base‘𝑎)
34 eqid 2825 . . . . . . . . . . . . . 14 (Base‘𝑏) = (Base‘𝑏)
351, 2, 3, 4, 5, 6, 7, 33, 34funcestrcsetclem6 17138 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36353expa 1151 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3736eqeq2d 2835 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837rexbidva 3259 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘))
3938adantr 474 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘))
4032, 39mpbird 249 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘))
41 eqid 2825 . . . . . . . . . . 11 (Hom ‘𝐸) = (Hom ‘𝐸)
421, 5estrcbas 17117 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐸))
4342, 3syl6reqr 2880 . . . . . . . . . . . . . . 15 (𝜑𝐵 = 𝑈)
4443eleq2d 2892 . . . . . . . . . . . . . 14 (𝜑 → (𝑎𝐵𝑎𝑈))
4544biimpcd 241 . . . . . . . . . . . . 13 (𝑎𝐵 → (𝜑𝑎𝑈))
4645adantr 474 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
4746impcom 398 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
4843eleq2d 2892 . . . . . . . . . . . . . 14 (𝜑 → (𝑏𝐵𝑏𝑈))
4948biimpcd 241 . . . . . . . . . . . . 13 (𝑏𝐵 → (𝜑𝑏𝑈))
5049adantl 475 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
5150impcom 398 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
521, 10, 41, 47, 51, 33, 34estrchom 17119 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
5352rexeqdv 3357 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5453adantr 474 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5540, 54mpbird 249 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
5655ex 403 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5722, 56sylbid 232 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5857ralrimiv 3174 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
59 dffo3 6623 . . . 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 578 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
6160ralrimivva 3180 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
623, 11, 41isfull2 16923 . 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 578 1 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118  Vcvv 3414   class class class wbr 4873  cmpt 4952   I cid 5249  cres 5344  wf 6119  ontowfo 6121  cfv 6123  (class class class)co 6905  cmpt2 6907  𝑚 cmap 8122  WUnicwun 9837  Basecbs 16222  Hom chom 16316   Func cfunc 16866   Full cful 16914  SetCatcsetc 17077  ExtStrCatcestrc 17114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-wun 9839  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-hom 16329  df-cco 16330  df-cat 16681  df-cid 16682  df-func 16870  df-full 16916  df-setc 17078  df-estrc 17115
This theorem is referenced by:  equivestrcsetc  17145
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