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Theorem fthestrcsetc 16613
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 16612 . 2 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 16610 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
105adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
11 eqid 2610 . . . . . . . . . . . . 13 (Hom ‘𝐸) = (Hom ‘𝐸)
121, 5estrcbas 16588 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘𝐸))
1312, 3syl6reqr 2663 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = 𝑈)
1413eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑎𝐵𝑎𝑈))
1514biimpcd 238 . . . . . . . . . . . . . . 15 (𝑎𝐵 → (𝜑𝑎𝑈))
1615adantr 480 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
1716impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
1813eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑏𝐵𝑏𝑈))
1918biimpcd 238 . . . . . . . . . . . . . . 15 (𝑏𝐵 → (𝜑𝑏𝑈))
2019adantl 481 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑏𝐵) → (𝜑𝑏𝑈))
2120impcom 445 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
22 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑎) = (Base‘𝑎)
23 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑏) = (Base‘𝑏)
241, 10, 11, 17, 21, 22, 23estrchom 16590 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝐸)𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2524eleq2d 2673 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16608 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘) = )
27263expia 1259 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘) = ))
2825, 27sylbid 229 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝑎𝐺𝑏)‘) = ))
2928com12 32 . . . . . . . . 9 ( ∈ (𝑎(Hom ‘𝐸)𝑏) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3029adantr 480 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘) = ))
3130impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘) = )
3224eleq2d 2673 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏) ↔ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 16608 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ 𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
34333expia 1259 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑘 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3532, 34sylbid 229 . . . . . . . . . 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 481 . . . . . . . 8 (( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)) → ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘))
3837impcom 445 . . . . . . 7 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → ((𝑎𝐺𝑏)‘𝑘) = 𝑘)
3931, 38eqeq12d 2625 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) ↔ = 𝑘))
4039biimpd 218 . . . . 5 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏))) → (((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
4140ralrimivva 2954 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀ ∈ (𝑎(Hom ‘𝐸)𝑏)∀𝑘 ∈ (𝑎(Hom ‘𝐸)𝑏)(((𝑎𝐺𝑏)‘) = ((𝑎𝐺𝑏)‘𝑘) → = 𝑘))
42 dff13 6416 . . . 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 695 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
4443ralrimivva 2954 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)–1-1→((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
45 eqid 2610 . . 3 (Hom ‘𝑆) = (Hom ‘𝑆)
463, 11, 45isfth2 16398 . 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 695 1 (𝜑𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4583  cmpt 4643   I cid 4948  cres 5040  wf 5800  1-1wf1 5801  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744  WUnicwun 9401  Basecbs 15695  Hom chom 15779   Func cfunc 16337   Faith cfth 16386  SetCatcsetc 16548  ExtStrCatcestrc 16585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-fth 16388  df-setc 16549  df-estrc 16586
This theorem is referenced by:  equivestrcsetc  16615
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