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Theorem fthestrcsetc 17375
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 17374 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 17372 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2820 . . . . . . . . . . . . 13 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 5estrcbas 17350 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐸))
1312, 3syl6reqr 2874 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = 𝑈)
1413eleq2d 2896 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐵𝑎𝑈))
1514biimpcd 251 . . . . . . . . . . . . . . 15 (𝑎𝐵 → (𝜑𝑎𝑈))
1615adantr 483 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
1716impcom 410 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
1813eleq2d 2896 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐵𝑏𝑈))
1918biimpcd 251 . . . . . . . . . . . . . . 15 (𝑏𝐵 → (𝜑𝑏𝑈))
2019adantl 484 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
2120impcom 410 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
22 eqid 2820 . . . . . . . . . . . . 13 (Base‘𝑎) = (Base‘𝑎)
23 eqid 2820 . . . . . . . . . . . . 13 (Base‘𝑏) = (Base‘𝑏)
241, 10, 11, 17, 21, 22, 23estrchom 17352 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2524eleq2d 2896 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 17370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘) = )
27263expia 1117 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘) = ))
2825, 27sylbid 242 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2928com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3029adantr 483 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3130impcom 410 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3224eleq2d 2896 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 17370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34333expia 1117 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3532, 34sylbid 242 . . . . . . . . . 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 484 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3837impcom 410 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3931, 38eqeq12d 2836 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4039biimpd 231 . . . . 5 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
4140ralrimivva 3178 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
42 dff13 6986 . . . 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 585 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
4443ralrimivva 3178 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
45 eqid 2820 . . 3 (Hom ‘𝑆) = (Hom ‘𝑆)
463, 11, 45isfth2 17160 . 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 585 1 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125   class class class wbr 5038  cmpt 5118   I cid 5431  cres 5529  wf 6323  1-1wf1 6324  cfv 6327  (class class class)co 7129  cmpo 7131  m cmap 8380  WUnicwun 10096  Basecbs 16458  Hom chom 16551   Func cfunc 17099   Faith cfth 17148  SetCatcsetc 17310  ExtStrCatcestrc 17347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568  ax-1cn 10569  ax-icn 10570  ax-addcl 10571  ax-addrcl 10572  ax-mulcl 10573  ax-mulrcl 10574  ax-mulcom 10575  ax-addass 10576  ax-mulass 10577  ax-distr 10578  ax-i2m1 10579  ax-1ne0 10580  ax-1rid 10581  ax-rnegex 10582  ax-rrecex 10583  ax-cnre 10584  ax-pre-lttri 10585  ax-pre-lttrn 10586  ax-pre-ltadd 10587  ax-pre-mulgt0 10588
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-int 4849  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-tr 5145  df-id 5432  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-mpo 7134  df-om 7555  df-1st 7663  df-2nd 7664  df-wrecs 7921  df-recs 7982  df-rdg 8020  df-1o 8076  df-oadd 8080  df-er 8263  df-map 8382  df-ixp 8436  df-en 8484  df-dom 8485  df-sdom 8486  df-fin 8487  df-wun 10098  df-pnf 10651  df-mnf 10652  df-xr 10653  df-ltxr 10654  df-le 10655  df-sub 10846  df-neg 10847  df-nn 11613  df-2 11675  df-3 11676  df-4 11677  df-5 11678  df-6 11679  df-7 11680  df-8 11681  df-9 11682  df-n0 11873  df-z 11957  df-dec 12074  df-uz 12219  df-fz 12873  df-struct 16460  df-ndx 16461  df-slot 16462  df-base 16464  df-hom 16564  df-cco 16565  df-cat 16914  df-cid 16915  df-func 17103  df-fth 17150  df-setc 17311  df-estrc 17348
This theorem is referenced by:  equivestrcsetc  17377
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