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Theorem dfrngc2 41737
Description: Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
Hypotheses
Ref Expression
dfrngc2.c 𝐶 = (RngCat‘𝑈)
dfrngc2.u (𝜑𝑈𝑉)
dfrngc2.b (𝜑𝐵 = (𝑈 ∩ Rng))
dfrngc2.h (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
dfrngc2.o (𝜑· = (comp‘(ExtStrCat‘𝑈)))
Assertion
Ref Expression
dfrngc2 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Proof of Theorem dfrngc2
Dummy variables 𝑥 𝑓 𝑔 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrngc2.c . . 3 𝐶 = (RngCat‘𝑈)
2 dfrngc2.u . . 3 (𝜑𝑈𝑉)
3 dfrngc2.b . . 3 (𝜑𝐵 = (𝑈 ∩ Rng))
4 dfrngc2.h . . 3 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
51, 2, 3, 4rngcval 41727 . 2 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6 eqid 2620 . . . 4 ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7 fvexd 6190 . . . 4 (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8 inex1g 4792 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
92, 8syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Rng) ∈ V)
103, 9eqeltrd 2699 . . . 4 (𝜑𝐵 ∈ V)
113, 4rnghmresfn 41728 . . . 4 (𝜑𝐻 Fn (𝐵 × 𝐵))
126, 7, 10, 11rescval2 16469 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13 eqid 2620 . . . . 5 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14 eqidd 2621 . . . . 5 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
15 dfrngc2.o . . . . . 6 (𝜑· = (comp‘(ExtStrCat‘𝑈)))
16 eqid 2620 . . . . . . 7 (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
1713, 2, 16estrccofval 16750 . . . . . 6 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1815, 17eqtrd 2654 . . . . 5 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1913, 2, 14, 18estrcval 16745 . . . 4 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
202, 2jca 554 . . . . 5 (𝜑 → (𝑈𝑉𝑈𝑉))
21 eqid 2620 . . . . . 6 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
2221mpt2exg 7230 . . . . 5 ((𝑈𝑉𝑈𝑉) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
2320, 22syl 17 . . . 4 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
24 fvexd 6190 . . . . 5 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
2515, 24eqeltrd 2699 . . . 4 (𝜑· ∈ V)
26 rnghmfn 41655 . . . . . . 7 RngHomo Fn (Rng × Rng)
27 fnfun 5976 . . . . . . 7 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
2826, 27mp1i 13 . . . . . 6 (𝜑 → Fun RngHomo )
29 sqxpexg 6948 . . . . . . 7 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
3010, 29syl 17 . . . . . 6 (𝜑 → (𝐵 × 𝐵) ∈ V)
31 resfunexg 6464 . . . . . 6 ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
3228, 30, 31syl2anc 692 . . . . 5 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
334, 32eqeltrd 2699 . . . 4 (𝜑𝐻 ∈ V)
34 inss1 3825 . . . . . 6 (𝑈 ∩ Rng) ⊆ 𝑈
3534a1i 11 . . . . 5 (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
363sseq1d 3624 . . . . 5 (𝜑 → (𝐵𝑈 ↔ (𝑈 ∩ Rng) ⊆ 𝑈))
3735, 36mpbird 247 . . . 4 (𝜑𝐵𝑈)
3819, 2, 23, 25, 10, 33, 37estrres 16760 . . 3 (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
3912, 38eqtrd 2654 . 2 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
405, 39eqtrd 2654 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  wss 3567  {ctp 4172  cop 4174   × cxp 5102  cres 5106  ccom 5108  Fun wfun 5870   Fn wfn 5871  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  𝑚 cmap 7842  ndxcnx 15835   sSet csts 15836  Basecbs 15838  s cress 15839  Hom chom 15933  compcco 15934  cat cresc 16449  ExtStrCatcestrc 16743  Rngcrng 41639   RngHomo crngh 41650  RngCatcrngc 41722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-hom 15947  df-cco 15948  df-resc 16452  df-estrc 16744  df-rnghomo 41652  df-rngc 41724
This theorem is referenced by:  rngcresringcat  41795
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