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Theorem dfringc2 41810
 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 41800 . 2 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6 eqid 2610 . . 3 ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7 fvex 6113 . . . 4 (ExtStrCat‘𝑈) ∈ V
87a1i 11 . . 3 (𝜑 → (ExtStrCat‘𝑈) ∈ V)
9 inex1g 4729 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
102, 9syl 17 . . . 4 (𝜑 → (𝑈 ∩ Ring) ∈ V)
113, 10eqeltrd 2688 . . 3 (𝜑𝐵 ∈ V)
123, 4rhmresfn 41801 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
136, 8, 11, 12rescval2 16311 . 2 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14 eqid 2610 . . . 4 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
15 eqidd 2611 . . . 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 2610 . . . . . 6 (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
1814, 2, 17estrccofval 16592 . . . . 5 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1916, 18eqtrd 2644 . . . 4 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
2014, 2, 15, 19estrcval 16587 . . 3 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
21 eqid 2610 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
2221mpt2exg 7134 . . . 4 ((𝑈𝑉𝑈𝑉) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
232, 2, 22syl2anc 691 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
24 fvex 6113 . . . . 5 (comp‘(ExtStrCat‘𝑈)) ∈ V
2524a1i 11 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
2616, 25eqeltrd 2688 . . 3 (𝜑· ∈ V)
27 rhmfn 41708 . . . . . 6 RingHom Fn (Ring × Ring)
28 fnfun 5902 . . . . . 6 ( RingHom Fn (Ring × Ring) → Fun RingHom )
2927, 28mp1i 13 . . . . 5 (𝜑 → Fun RingHom )
30 sqxpexg 6861 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
3111, 30syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
32 resfunexg 6384 . . . . 5 ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
3329, 31, 32syl2anc 691 . . . 4 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V)
344, 33eqeltrd 2688 . . 3 (𝜑𝐻 ∈ V)
35 inss1 3795 . . . 4 (𝑈 ∩ Ring) ⊆ 𝑈
363, 35syl6eqss 3618 . . 3 (𝜑𝐵𝑈)
3720, 2, 23, 26, 11, 34, 36estrres 16602 . 2 (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
385, 13, 373eqtrd 2648 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  {ctp 4129  ⟨cop 4131   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾s cress 15696  Hom chom 15779  compcco 15780   ↾cat cresc 16291  ExtStrCatcestrc 16585  Ringcrg 18370   RingHom crh 18535  RingCatcringc 41795 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-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-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-resc 16294  df-estrc 16586  df-mhm 17158  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-rnghom 18538  df-ringc 41797 This theorem is referenced by:  rngcresringcat  41822
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