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Theorem fthestrcsetc 16706
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 16705 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 16703 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2626 . . . . . . . . . . . . 13 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 5estrcbas 16681 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐸))
1312, 3syl6reqr 2679 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = 𝑈)
1413eleq2d 2689 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐵𝑎𝑈))
1514biimpcd 239 . . . . . . . . . . . . . . 15 (𝑎𝐵 → (𝜑𝑎𝑈))
1615adantr 481 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
1716impcom 446 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
1813eleq2d 2689 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐵𝑏𝑈))
1918biimpcd 239 . . . . . . . . . . . . . . 15 (𝑏𝐵 → (𝜑𝑏𝑈))
2019adantl 482 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
2120impcom 446 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
22 eqid 2626 . . . . . . . . . . . . 13 (Base‘𝑎) = (Base‘𝑎)
23 eqid 2626 . . . . . . . . . . . . 13 (Base‘𝑏) = (Base‘𝑏)
241, 10, 11, 17, 21, 22, 23estrchom 16683 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2524eleq2d 2689 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16701 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘) = )
27263expia 1264 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘) = ))
2825, 27sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2928com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3029adantr 481 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3130impcom 446 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3224eleq2d 2689 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16701 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34333expia 1264 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3532, 34sylbid 230 . . . . . . . . . 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 482 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3837impcom 446 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3931, 38eqeq12d 2641 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4039biimpd 219 . . . . 5 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
4140ralrimivva 2970 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
42 dff13 6467 . . . 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 697 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
4443ralrimivva 2970 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
45 eqid 2626 . . 3 (Hom ‘𝑆) = (Hom ‘𝑆)
463, 11, 45isfth2 16491 . 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 697 1 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912   class class class wbr 4618  cmpt 4678   I cid 4989  cres 5081  wf 5846  1-1wf1 5847  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑚 cmap 7803  WUnicwun 9467  Basecbs 15776  Hom chom 15868   Func cfunc 16430   Faith cfth 16479  SetCatcsetc 16641  ExtStrCatcestrc 16678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-wun 9469  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-hom 15882  df-cco 15883  df-cat 16245  df-cid 16246  df-func 16434  df-fth 16481  df-setc 16642  df-estrc 16679
This theorem is referenced by:  equivestrcsetc  16708
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