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Theorem rhmsscmap2 20554
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rhmsscmap.u (𝜑𝑈𝑉)
rhmsscmap.r (𝜑𝑅 = (Ring ∩ 𝑈))
Assertion
Ref Expression
rhmsscmap2 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Distinct variable group:   𝑥,𝑅,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssidd 4000 . 2 (𝜑𝑅𝑅)
2 eqid 2726 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
3 eqid 2726 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
42, 3rhmf 20387 . . . . . 6 ( ∈ (𝑎 RingHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
5 simpr 484 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
6 fvex 6898 . . . . . . . . . 10 (Base‘𝑏) ∈ V
7 fvex 6898 . . . . . . . . . 10 (Base‘𝑎) ∈ V
86, 7pm3.2i 470 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9 elmapg 8835 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
108, 9mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
115, 10mpbird 257 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
1211ex 412 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
134, 12syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RingHom 𝑏) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
1413ssrdv 3983 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
15 ovres 7570 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
1615adantl 481 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
17 eqidd 2727 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
18 fveq2 6885 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19 fveq2 6885 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2018, 19oveqan12rd 7425 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2120adantl 481 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
22 simpl 482 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
23 simpr 484 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
24 ovexd 7440 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
2517, 21, 22, 23, 24ovmpod 7556 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2625adantl 481 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2714, 16, 263sstr4d 4024 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
2827ralrimivva 3194 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
29 rhmfn 20401 . . . . 5 RingHom Fn (Ring × Ring)
3029a1i 11 . . . 4 (𝜑 → RingHom Fn (Ring × Ring))
31 rhmsscmap.r . . . . . 6 (𝜑𝑅 = (Ring ∩ 𝑈))
32 inss1 4223 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
3331, 32eqsstrdi 4031 . . . . 5 (𝜑𝑅 ⊆ Ring)
34 xpss12 5684 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
3533, 33, 34syl2anc 583 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
36 fnssres 6667 . . . 4 (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3730, 35, 36syl2anc 583 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38 eqid 2726 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
39 ovex 7438 . . . . 5 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
4038, 39fnmpoi 8055 . . . 4 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅)
4140a1i 11 . . 3 (𝜑 → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅))
42 incom 4196 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
43 rhmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
44 inex1g 5312 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4543, 44syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Ring) ∈ V)
4642, 45eqeltrid 2831 . . . 4 (𝜑 → (Ring ∩ 𝑈) ∈ V)
4731, 46eqeltrd 2827 . . 3 (𝜑𝑅 ∈ V)
4837, 41, 47isssc 17776 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅𝑅 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))))
491, 28, 48mpbir2and 710 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  Vcvv 3468  cin 3942  wss 3943   class class class wbr 5141   × cxp 5667  cres 5671   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405  cmpo 7407  m cmap 8822  Basecbs 17153  cat cssc 17763  Ringcrg 20138   RingHom crh 20371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-plusg 17219  df-0g 17396  df-ssc 17766  df-mhm 18713  df-ghm 19139  df-mgp 20040  df-ur 20087  df-ring 20140  df-rhm 20374
This theorem is referenced by: (None)
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