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Theorem funcestrcsetc 16560
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, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-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
funcestrcsetc (𝜑𝐹(𝐸 Func 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 funcestrcsetc
Dummy variables 𝑎 𝑏 𝑐 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcestrcsetc.b . 2 𝐵 = (Base‘𝐸)
2 funcestrcsetc.c . 2 𝐶 = (Base‘𝑆)
3 eqid 2609 . 2 (Hom ‘𝐸) = (Hom ‘𝐸)
4 eqid 2609 . 2 (Hom ‘𝑆) = (Hom ‘𝑆)
5 eqid 2609 . 2 (Id‘𝐸) = (Id‘𝐸)
6 eqid 2609 . 2 (Id‘𝑆) = (Id‘𝑆)
7 eqid 2609 . 2 (comp‘𝐸) = (comp‘𝐸)
8 eqid 2609 . 2 (comp‘𝑆) = (comp‘𝑆)
9 funcestrcsetc.u . . 3 (𝜑𝑈 ∈ WUni)
10 funcestrcsetc.e . . . 4 𝐸 = (ExtStrCat‘𝑈)
1110estrccat 16544 . . 3 (𝑈 ∈ WUni → 𝐸 ∈ Cat)
129, 11syl 17 . 2 (𝜑𝐸 ∈ Cat)
13 funcestrcsetc.s . . . 4 𝑆 = (SetCat‘𝑈)
1413setccat 16506 . . 3 (𝑈 ∈ WUni → 𝑆 ∈ Cat)
159, 14syl 17 . 2 (𝜑𝑆 ∈ Cat)
16 funcestrcsetc.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
1710, 13, 1, 2, 9, 16funcestrcsetclem3 16553 . 2 (𝜑𝐹:𝐵𝐶)
18 funcestrcsetc.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
1910, 13, 1, 2, 9, 16, 18funcestrcsetclem4 16554 . 2 (𝜑𝐺 Fn (𝐵 × 𝐵))
2010, 13, 1, 2, 9, 16, 18funcestrcsetclem8 16558 . 2 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝐸)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
2110, 13, 1, 2, 9, 16, 18funcestrcsetclem7 16557 . 2 ((𝜑𝑎𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝐸)‘𝑎)) = ((Id‘𝑆)‘(𝐹𝑎)))
2210, 13, 1, 2, 9, 16, 18funcestrcsetclem9 16559 . 2 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵) ∧ ( ∈ (𝑎(Hom ‘𝐸)𝑏) ∧ 𝑘 ∈ (𝑏(Hom ‘𝐸)𝑐))) → ((𝑎𝐺𝑐)‘(𝑘(⟨𝑎, 𝑏⟩(comp‘𝐸)𝑐))) = (((𝑏𝐺𝑐)‘𝑘)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝑆)(𝐹𝑐))((𝑎𝐺𝑏)‘)))
231, 2, 3, 4, 5, 6, 7, 8, 12, 15, 17, 19, 20, 21, 22isfuncd 16296 1 (𝜑𝐹(𝐸 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976   class class class wbr 4577  cmpt 4637   I cid 4937  cres 5029  cfv 5789  (class class class)co 6526  cmpt2 6528  𝑚 cmap 7721  WUnicwun 9378  Basecbs 15643  Hom chom 15727  compcco 15728  Catccat 16096  Idccid 16097   Func cfunc 16285  SetCatcsetc 16496  ExtStrCatcestrc 16533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-wun 9380  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-fz 12155  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-hom 15741  df-cco 15742  df-cat 16100  df-cid 16101  df-func 16289  df-setc 16497  df-estrc 16534
This theorem is referenced by:  fthestrcsetc  16561  fullestrcsetc  16562  funcrngcsetc  41771  funcrngcsetcALT  41772  funcringcsetc  41808
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