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Theorem dfringc2 41809
Description: Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
Hypotheses
Ref Expression
dfringc2.c 𝐶 = (RingCat‘𝑈)
dfringc2.u (𝜑𝑈𝑉)
dfringc2.b (𝜑𝐵 = (𝑈 ∩ Ring))
dfringc2.h (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
dfringc2.o (𝜑· = (comp‘(ExtStrCat‘𝑈)))
Assertion
Ref Expression
dfringc2 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Proof of Theorem dfringc2
Dummy variables 𝑓 𝑔 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfringc2.c . . 3 𝐶 = (RingCat‘𝑈)
2 dfringc2.u . . 3 (𝜑𝑈𝑉)
3 dfringc2.b . . 3 (𝜑𝐵 = (𝑈 ∩ Ring))
4 dfringc2.h . . 3 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
51, 2, 3, 4ringcval 41799 . 2 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6 eqid 2606 . . 3 ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7 fvex 6095 . . . 4 (ExtStrCat‘𝑈) ∈ V
87a1i 11 . . 3 (𝜑 → (ExtStrCat‘𝑈) ∈ V)
9 inex1g 4721 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
102, 9syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
113, 10eqeltrd 2684 . . 3 (𝜑𝐵 ∈ V)
123, 4rhmresfn 41800 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
136, 8, 11, 12rescval2 16254 . 2 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14 eqid 2606 . . . 4 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
15 eqidd 2607 . . . 4 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
16 dfringc2.o . . . . 5 (𝜑· = (comp‘(ExtStrCat‘𝑈)))
17 eqid 2606 . . . . . 6 (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
1814, 2, 17estrccofval 16535 . . . . 5 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1916, 18eqtrd 2640 . . . 4 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
2014, 2, 15, 19estrcval 16530 . . 3 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
21 eqid 2606 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
2221mpt2exg 7108 . . . 4 ((𝑈𝑉𝑈𝑉) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
232, 2, 22syl2anc 690 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
24 fvex 6095 . . . . 5 (comp‘(ExtStrCat‘𝑈)) ∈ V
2524a1i 11 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
2616, 25eqeltrd 2684 . . 3 (𝜑· ∈ V)
27 rhmfn 41707 . . . . . 6 RingHom Fn (Ring × Ring)
28 fnfun 5885 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2927, 28mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
30 sqxpexg 6835 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
3111, 30syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
32 resfunexg 6359 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3329, 31, 32syl2anc 690 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
344, 33eqeltrd 2684 . . 3 (𝜑𝐻 ∈ V)
35 inss1 3791 . . . 4 (𝑈 ∩ Ring) ⊆ 𝑈
363, 35syl6eqss 3614 . . 3 (𝜑𝐵𝑈)
3720, 2, 23, 26, 11, 34, 36estrres 16545 . 2 (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
385, 13, 373eqtrd 2644 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3169  cin 3535  {ctp 4125  cop 4127   × cxp 5023  cres 5027  ccom 5029  Fun wfun 5781   Fn wfn 5782  cfv 5787  (class class class)co 6524  cmpt2 6526  1st c1st 7031  2nd c2nd 7032  𝑚 cmap 7718  ndxcnx 15635   sSet csts 15636  Basecbs 15638  s cress 15639  Hom chom 15722  compcco 15723  cat cresc 16234  ExtStrCatcestrc 16528  Ringcrg 18313   RingHom crh 18478  RingCatcringc 41794
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-fz 12150  df-struct 15640  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-ress 15645  df-plusg 15724  df-hom 15736  df-cco 15737  df-0g 15868  df-resc 16237  df-estrc 16529  df-mhm 17101  df-ghm 17424  df-mgp 18256  df-ur 18268  df-ring 18315  df-rnghom 18481  df-ringc 41796
This theorem is referenced by:  rngcresringcat  41821
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