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Theorem rngcrescrhmALTV 41918
Description: The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcrescrhmALTV.u (𝜑𝑈𝑉)
rngcrescrhmALTV.c 𝐶 = (RngCatALTV‘𝑈)
rngcrescrhmALTV.r (𝜑𝑅 = (Ring ∩ 𝑈))
rngcrescrhmALTV.h 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
Assertion
Ref Expression
rngcrescrhmALTV (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Proof of Theorem rngcrescrhmALTV
StepHypRef Expression
1 eqid 2609 . 2 (𝐶cat 𝐻) = (𝐶cat 𝐻)
2 rngcrescrhmALTV.c . . . 4 𝐶 = (RngCatALTV‘𝑈)
3 fvex 6098 . . . 4 (RngCatALTV‘𝑈) ∈ V
42, 3eqeltri 2683 . . 3 𝐶 ∈ V
54a1i 11 . 2 (𝜑𝐶 ∈ V)
6 rngcrescrhmALTV.r . . . 4 (𝜑𝑅 = (Ring ∩ 𝑈))
7 incom 3766 . . . 4 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
86, 7syl6eq 2659 . . 3 (𝜑𝑅 = (𝑈 ∩ Ring))
9 rngcrescrhmALTV.u . . . 4 (𝜑𝑈𝑉)
10 inex1g 4724 . . . 4 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
119, 10syl 17 . . 3 (𝜑 → (𝑈 ∩ Ring) ∈ V)
128, 11eqeltrd 2687 . 2 (𝜑𝑅 ∈ V)
13 inss1 3794 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
146, 13syl6eqss 3617 . . . . 5 (𝜑𝑅 ⊆ Ring)
15 xpss12 5137 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
1614, 14, 15syl2anc 690 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
17 rhmfn 41730 . . . . 5 RingHom Fn (Ring × Ring)
18 fnssresb 5903 . . . . 5 ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
1917, 18mp1i 13 . . . 4 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
2016, 19mpbird 245 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
21 rngcrescrhmALTV.h . . . 4 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
2221fneq1i 5885 . . 3 (𝐻 Fn (𝑅 × 𝑅) ↔ ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
2320, 22sylibr 222 . 2 (𝜑𝐻 Fn (𝑅 × 𝑅))
241, 5, 12, 23rescval2 16260 1 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  wss 3539  cop 4130   × cxp 5026  cres 5030   Fn wfn 5785  cfv 5790  (class class class)co 6527  ndxcnx 15641   sSet csts 15642  s cress 15645  Hom chom 15728  cat cresc 16240  Ringcrg 18319   RingHom crh 18484  RngCatALTVcrngcALTV 41772
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 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
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 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-plusg 15730  df-0g 15874  df-resc 16243  df-mhm 17107  df-ghm 17430  df-mgp 18262  df-ur 18274  df-ring 18321  df-rnghom 18487
This theorem is referenced by: (None)
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