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Theorem dfrngc2 44388
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 𝑓 𝑔 𝑣 𝑥 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 44378 . 2 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6 eqid 2821 . . 3 ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7 fvexd 6658 . . 3 (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8 inex1g 5196 . . . . 5 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
92, 8syl 17 . . . 4 (𝜑 → (𝑈 ∩ Rng) ∈ V)
103, 9eqeltrd 2912 . . 3 (𝜑𝐵 ∈ V)
113, 4rnghmresfn 44379 . . 3 (𝜑𝐻 Fn (𝐵 × 𝐵))
126, 7, 10, 11rescval2 17076 . 2 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13 eqid 2821 . . . 4 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14 eqidd 2822 . . . 4 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
15 dfrngc2.o . . . . 5 (𝜑· = (comp‘(ExtStrCat‘𝑈)))
16 eqid 2821 . . . . . 6 (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
1713, 2, 16estrccofval 17357 . . . . 5 (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1815, 17eqtrd 2856 . . . 4 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1913, 2, 14, 18estrcval 17352 . . 3 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20 mpoexga 7750 . . . 4 ((𝑈𝑉𝑈𝑉) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
212, 2, 20syl2anc 587 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
22 fvexd 6658 . . . 4 (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
2315, 22eqeltrd 2912 . . 3 (𝜑· ∈ V)
24 rnghmfn 44306 . . . . . 6 RngHomo Fn (Rng × Rng)
25 fnfun 6426 . . . . . 6 ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
2624, 25mp1i 13 . . . . 5 (𝜑 → Fun RngHomo )
27 sqxpexg 7452 . . . . . 6 (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
2810, 27syl 17 . . . . 5 (𝜑 → (𝐵 × 𝐵) ∈ V)
29 resfunexg 6951 . . . . 5 ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
3026, 28, 29syl2anc 587 . . . 4 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
314, 30eqeltrd 2912 . . 3 (𝜑𝐻 ∈ V)
32 inss1 4180 . . . 4 (𝑈 ∩ Rng) ⊆ 𝑈
333, 32eqsstrdi 3997 . . 3 (𝜑𝐵𝑈)
3419, 2, 21, 23, 31, 33estrres 17367 . 2 (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
355, 12, 343eqtrd 2860 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3471  cin 3909  {ctp 4544  cop 4546   × cxp 5526  cres 5530  ccom 5532  Fun wfun 6322   Fn wfn 6323  cfv 6328  (class class class)co 7130  cmpo 7132  1st c1st 7662  2nd c2nd 7663  m cmap 8381  ndxcnx 16458   sSet csts 16459  Basecbs 16461  s cress 16462  Hom chom 16554  compcco 16555  cat cresc 17056  ExtStrCatcestrc 17350  Rngcrng 44290   RngHomo crngh 44301  RngCatcrngc 44373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-fz 12876  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-hom 16567  df-cco 16568  df-resc 17059  df-estrc 17351  df-rnghomo 44303  df-rngc 44375
This theorem is referenced by:  rngcresringcat  44446
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