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Theorem funcringcsetcALTV 44237
Description: The "natural forgetful functor" from the category of rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetcALTV (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetcALTV
Dummy variables 𝑎 𝑏 𝑐 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcringcsetcALTV.b . 2 𝐵 = (Base‘𝑅)
2 funcringcsetcALTV.c . 2 𝐶 = (Base‘𝑆)
3 eqid 2821 . 2 (Hom ‘𝑅) = (Hom ‘𝑅)
4 eqid 2821 . 2 (Hom ‘𝑆) = (Hom ‘𝑆)
5 eqid 2821 . 2 (Id‘𝑅) = (Id‘𝑅)
6 eqid 2821 . 2 (Id‘𝑆) = (Id‘𝑆)
7 eqid 2821 . 2 (comp‘𝑅) = (comp‘𝑅)
8 eqid 2821 . 2 (comp‘𝑆) = (comp‘𝑆)
9 funcringcsetcALTV.u . . 3 (𝜑𝑈 ∈ WUni)
10 funcringcsetcALTV.r . . . 4 𝑅 = (RingCatALTV‘𝑈)
1110ringccatALTV 44222 . . 3 (𝑈 ∈ WUni → 𝑅 ∈ Cat)
129, 11syl 17 . 2 (𝜑𝑅 ∈ Cat)
13 funcringcsetcALTV.s . . . 4 𝑆 = (SetCat‘𝑈)
1413setccat 17335 . . 3 (𝑈 ∈ WUni → 𝑆 ∈ Cat)
159, 14syl 17 . 2 (𝜑𝑆 ∈ Cat)
16 funcringcsetcALTV.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
1710, 13, 1, 2, 9, 16funcringcsetclem3ALTV 44230 . 2 (𝜑𝐹:𝐵𝐶)
18 funcringcsetcALTV.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
1910, 13, 1, 2, 9, 16, 18funcringcsetclem4ALTV 44231 . 2 (𝜑𝐺 Fn (𝐵 × 𝐵))
2010, 13, 1, 2, 9, 16, 18funcringcsetclem8ALTV 44235 . 2 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎𝐺𝑏):(𝑎(Hom ‘𝑅)𝑏)⟶((𝐹𝑎)(Hom ‘𝑆)(𝐹𝑏)))
2110, 13, 1, 2, 9, 16, 18funcringcsetclem7ALTV 44234 . 2 ((𝜑𝑎𝐵) → ((𝑎𝐺𝑎)‘((Id‘𝑅)‘𝑎)) = ((Id‘𝑆)‘(𝐹𝑎)))
2210, 13, 1, 2, 9, 16, 18funcringcsetclem9ALTV 44236 . 2 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵) ∧ ( ∈ (𝑎(Hom ‘𝑅)𝑏) ∧ 𝑘 ∈ (𝑏(Hom ‘𝑅)𝑐))) → ((𝑎𝐺𝑐)‘(𝑘(⟨𝑎, 𝑏⟩(comp‘𝑅)𝑐))) = (((𝑏𝐺𝑐)‘𝑘)(⟨(𝐹𝑎), (𝐹𝑏)⟩(comp‘𝑆)(𝐹𝑐))((𝑎𝐺𝑏)‘)))
231, 2, 3, 4, 5, 6, 7, 8, 12, 15, 17, 19, 20, 21, 22isfuncd 17125 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105   class class class wbr 5058  cmpt 5138   I cid 5453  cres 5551  cfv 6349  (class class class)co 7145  cmpo 7147  WUnicwun 10111  Basecbs 16473  Hom chom 16566  compcco 16567  Catccat 16925  Idccid 16926   Func cfunc 17114  SetCatcsetc 17325   RingHom crh 19395  RingCatALTVcringcALTV 44173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-wun 10113  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-hom 16579  df-cco 16580  df-0g 16705  df-cat 16929  df-cid 16930  df-func 17118  df-setc 17326  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-mhm 17946  df-grp 18046  df-ghm 18296  df-mgp 19171  df-ur 19183  df-ring 19230  df-rnghom 19398  df-ringcALTV 44175
This theorem is referenced by: (None)
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