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Theorem dfringc2 41336
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 41326 . 2 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6 eqid 2621 . . 3 ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7 fvexd 6170 . . 3 (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8 inex1g 4771 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
92, 8syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
103, 9eqeltrd 2698 . . 3 (𝜑𝐵 ∈ V)
113, 4rhmresfn 41327 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
126, 7, 10, 11rescval2 16428 . 2 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13 eqid 2621 . . . 4 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14 eqidd 2622 . . . 4 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
15 dfringc2.o . . . . 5 (𝜑· = (comp‘(ExtStrCat‘𝑈)))
16 eqid 2621 . . . . . 6 (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
1713, 2, 16estrccofval 16709 . . . . 5 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1815, 17eqtrd 2655 . . . 4 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1913, 2, 14, 18estrcval 16704 . . 3 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20 eqid 2621 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
2120mpt2exg 7205 . . . 4 ((𝑈𝑉𝑈𝑉) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
222, 2, 21syl2anc 692 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
23 fvexd 6170 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
2415, 23eqeltrd 2698 . . 3 (𝜑· ∈ V)
25 rhmfn 41236 . . . . . 6 RingHom Fn (Ring × Ring)
26 fnfun 5956 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2725, 26mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
28 sqxpexg 6928 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2910, 28syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
30 resfunexg 6444 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3127, 29, 30syl2anc 692 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
324, 31eqeltrd 2698 . . 3 (𝜑𝐻 ∈ V)
33 inss1 3817 . . . 4 (𝑈 ∩ Ring) ⊆ 𝑈
343, 33syl6eqss 3640 . . 3 (𝜑𝐵𝑈)
3519, 2, 22, 24, 10, 32, 34estrres 16719 . 2 (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
365, 12, 353eqtrd 2659 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cin 3559  {ctp 4159  cop 4161   × cxp 5082  cres 5086  ccom 5088  Fun wfun 5851   Fn wfn 5852  cfv 5857  (class class class)co 6615  cmpt2 6617  1st c1st 7126  2nd c2nd 7127  𝑚 cmap 7817  ndxcnx 15797   sSet csts 15798  Basecbs 15800  s cress 15801  Hom chom 15892  compcco 15893  cat cresc 16408  ExtStrCatcestrc 16702  Ringcrg 18487   RingHom crh 18652  RingCatcringc 41321
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-hom 15906  df-cco 15907  df-0g 16042  df-resc 16411  df-estrc 16703  df-mhm 17275  df-ghm 17598  df-mgp 18430  df-ur 18442  df-ring 18489  df-rnghom 18655  df-ringc 41323
This theorem is referenced by:  rngcresringcat  41348
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