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Theorem rhmsscmap 41811
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
Hypotheses
Ref Expression
rhmsscmap.u (𝜑𝑈𝑉)
rhmsscmap.r (𝜑𝑅 = (Ring ∩ 𝑈))
Assertion
Ref Expression
rhmsscmap (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmsscmap.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
2 inss2 3792 . . 3 (Ring ∩ 𝑈) ⊆ 𝑈
31, 2syl6eqss 3614 . 2 (𝜑𝑅𝑈)
4 eqid 2606 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
5 eqid 2606 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
64, 5rhmf 18492 . . . . . 6 ( ∈ (𝑎 RingHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
7 simpr 475 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
8 fvex 6095 . . . . . . . . . 10 (Base‘𝑏) ∈ V
9 fvex 6095 . . . . . . . . . 10 (Base‘𝑎) ∈ V
108, 9pm3.2i 469 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11 elmapg 7731 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
1210, 11mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
137, 12mpbird 245 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1413ex 448 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
156, 14syl5 33 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RingHom 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1615ssrdv 3570 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
17 ovres 6673 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
1817adantl 480 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
19 eqidd 2607 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
20 fveq2 6085 . . . . . . 7 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21 fveq2 6085 . . . . . . 7 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2220, 21oveqan12rd 6544 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2322adantl 480 . . . . 5 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
243sseld 3563 . . . . . . . 8 (𝜑 → (𝑎𝑅𝑎𝑈))
2524com12 32 . . . . . . 7 (𝑎𝑅 → (𝜑𝑎𝑈))
2625adantr 479 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑎𝑈))
2726impcom 444 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑎𝑈)
283sseld 3563 . . . . . . . 8 (𝜑 → (𝑏𝑅𝑏𝑈))
2928com12 32 . . . . . . 7 (𝑏𝑅 → (𝜑𝑏𝑈))
3029adantl 480 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑏𝑈))
3130impcom 444 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑏𝑈)
32 ovex 6552 . . . . . 6 ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V
3332a1i 11 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
3419, 23, 27, 31, 33ovmpt2d 6661 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
3516, 18, 343sstr4d 3607 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
3635ralrimivva 2950 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
37 rhmfn 41707 . . . . 5 RingHom Fn (Ring × Ring)
3837a1i 11 . . . 4 (𝜑 → RingHom Fn (Ring × Ring))
39 inss1 3791 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
401, 39syl6eqss 3614 . . . . 5 (𝜑𝑅 ⊆ Ring)
41 xpss12 5134 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
4240, 40, 41syl2anc 690 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
43 fnssres 5901 . . . 4 (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
4438, 42, 43syl2anc 690 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
45 eqid 2606 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
46 ovex 6552 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4745, 46fnmpt2i 7102 . . . 4 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈)
4847a1i 11 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈))
49 rhmsscmap.u . . . 4 (𝜑𝑈𝑉)
50 elex 3181 . . . 4 (𝑈𝑉𝑈 ∈ V)
5149, 50syl 17 . . 3 (𝜑𝑈 ∈ V)
5244, 48, 51isssc 16246 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑈 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
533, 36, 52mpbir2and 958 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  Vcvv 3169  cin 3535  wss 3536   class class class wbr 4574   × cxp 5023  cres 5027   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  cmpt2 6526  𝑚 cmap 7718  Basecbs 15638  cat cssc 16233  Ringcrg 18313   RingHom crh 18478
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-map 7720  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-plusg 15724  df-0g 15868  df-ssc 16236  df-mhm 17101  df-ghm 17424  df-mgp 18256  df-ur 18268  df-ring 18315  df-rnghom 18481
This theorem is referenced by:  rhmsubcsetc  41814
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