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Theorem fullestrcsetc 16984
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 16982 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 16980 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2752 . . . . . . . 8 (Hom ‘𝑆) = (Hom ‘𝑆)
121, 2, 3, 4, 5, 6funcestrcsetclem2 16974 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) ∈ 𝑈)
1312adantrr 755 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) ∈ 𝑈)
141, 2, 3, 4, 5, 6funcestrcsetclem2 16974 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ 𝑈)
1514adantrl 754 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) ∈ 𝑈)
162, 10, 11, 13, 15elsetchom 16924 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(𝐹𝑎)⟶(𝐹𝑏)))
171, 2, 3, 4, 5, 6funcestrcsetclem1 16973 . . . . . . . . 9 ((𝜑𝑎𝐵) → (𝐹𝑎) = (Base‘𝑎))
1817adantrr 755 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) = (Base‘𝑎))
191, 2, 3, 4, 5, 6funcestrcsetclem1 16973 . . . . . . . . 9 ((𝜑𝑏𝐵) → (𝐹𝑏) = (Base‘𝑏))
2019adantrl 754 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) = (Base‘𝑏))
2118, 20feq23d 6193 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(𝐹𝑎)⟶(𝐹𝑏) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
2216, 21bitrd 268 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
23 fvex 6354 . . . . . . . . . . . . 13 (Base‘𝑏) ∈ V
24 fvex 6354 . . . . . . . . . . . . 13 (Base‘𝑎) ∈ V
2523, 24pm3.2i 470 . . . . . . . . . . . 12 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
26 elmapg 8028 . . . . . . . . . . . 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 503 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
29 equequ2 2100 . . . . . . . . . . 11 (𝑘 = → ( = 𝑘 = ))
3029adantl 473 . . . . . . . . . 10 ((((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) ∧ 𝑘 = ) → ( = 𝑘 = ))
31 eqidd 2753 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → = )
3228, 30, 31rspcedvd 3448 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘)
33 eqid 2752 . . . . . . . . . . . . . 14 (Base‘𝑎) = (Base‘𝑎)
34 eqid 2752 . . . . . . . . . . . . . 14 (Base‘𝑏) = (Base‘𝑏)
351, 2, 3, 4, 5, 6, 7, 33, 34funcestrcsetclem6 16978 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
36353expa 1111 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3736eqeq2d 2762 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ( = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
3837rexbidva 3179 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘))
3938adantr 472 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = 𝑘))
4032, 39mpbird 247 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘))
41 eqid 2752 . . . . . . . . . . 11 (Hom ‘𝐸) = (Hom ‘𝐸)
421, 5estrcbas 16958 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐸))
4342, 3syl6reqr 2805 . . . . . . . . . . . . . . 15 (𝜑𝐵 = 𝑈)
4443eleq2d 2817 . . . . . . . . . . . . . 14 (𝜑 → (𝑎𝐵𝑎𝑈))
4544biimpcd 239 . . . . . . . . . . . . 13 (𝑎𝐵 → (𝜑𝑎𝑈))
4645adantr 472 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
4746impcom 445 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
4843eleq2d 2817 . . . . . . . . . . . . . 14 (𝜑 → (𝑏𝐵𝑏𝑈))
4948biimpcd 239 . . . . . . . . . . . . 13 (𝑏𝐵 → (𝜑𝑏𝑈))
5049adantl 473 . . . . . . . . . . . 12 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
5150impcom 445 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
521, 10, 41, 47, 51, 33, 34estrchom 16960 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
5352rexeqdv 3276 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5453adantr 472 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → (∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘) ↔ ∃𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) = ((𝑎𝐺𝑏)‘𝑘)))
5540, 54mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
5655ex 449 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5722, 56sylbid 230 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)) → ∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘)))
5857ralrimiv 3095 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ ((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏))∃𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) = ((𝑎𝐺𝑏)‘𝑘))
59 dffo3 6529 . . . 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 701 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
6160ralrimivva 3101 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–onto→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
623, 11, 41isfull2 16764 . 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 701 1 (𝜑𝐹(𝐸 Full 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  Vcvv 3332   class class class wbr 4796  cmpt 4873   I cid 5165  cres 5260  wf 6037  ontowfo 6039  cfv 6041  (class class class)co 6805  cmpt2 6807  𝑚 cmap 8015  WUnicwun 9706  Basecbs 16051  Hom chom 16146   Func cfunc 16707   Full cful 16755  SetCatcsetc 16918  ExtStrCatcestrc 16955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-wun 9708  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-cat 16522  df-cid 16523  df-func 16711  df-full 16757  df-setc 16919  df-estrc 16956
This theorem is referenced by:  equivestrcsetc  16985
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