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Theorem rhmsscmap2 46917
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 4006 . 2 (𝜑𝑅𝑅)
2 eqid 2733 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
3 eqid 2733 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
42, 3rhmf 20263 . . . . . 6 ( ∈ (𝑎 RingHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
5 simpr 486 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
6 fvex 6905 . . . . . . . . . 10 (Base‘𝑏) ∈ V
7 fvex 6905 . . . . . . . . . 10 (Base‘𝑎) ∈ V
86, 7pm3.2i 472 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9 elmapg 8833 . . . . . . . . 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 414 . . . . . 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 3989 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
15 ovres 7573 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
1615adantl 483 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
17 eqidd 2734 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
18 fveq2 6892 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19 fveq2 6892 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2018, 19oveqan12rd 7429 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2120adantl 483 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
22 simpl 484 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
23 simpr 486 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
24 ovexd 7444 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
2517, 21, 22, 23, 24ovmpod 7560 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2625adantl 483 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2714, 16, 263sstr4d 4030 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
2827ralrimivva 3201 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
29 rhmfn 46721 . . . . 5 RingHom Fn (Ring × Ring)
3029a1i 11 . . . 4 (𝜑 → RingHom Fn (Ring × Ring))
31 rhmsscmap.r . . . . . 6 (𝜑𝑅 = (Ring ∩ 𝑈))
32 inss1 4229 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
3331, 32eqsstrdi 4037 . . . . 5 (𝜑𝑅 ⊆ Ring)
34 xpss12 5692 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
3533, 33, 34syl2anc 585 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
36 fnssres 6674 . . . 4 (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3730, 35, 36syl2anc 585 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38 eqid 2733 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
39 ovex 7442 . . . . 5 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
4038, 39fnmpoi 8056 . . . 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 4202 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
43 rhmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
44 inex1g 5320 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4543, 44syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Ring) ∈ V)
4642, 45eqeltrid 2838 . . . 4 (𝜑 → (Ring ∩ 𝑈) ∈ V)
4731, 46eqeltrd 2834 . . 3 (𝜑𝑅 ∈ V)
4837, 41, 47isssc 17767 . 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 712 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 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cin 3948  wss 3949   class class class wbr 5149   × cxp 5675  cres 5679   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  cmpo 7411  m cmap 8820  Basecbs 17144  cat cssc 17754  Ringcrg 20056   RingHom crh 20248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-ssc 17757  df-mhm 18671  df-ghm 19090  df-mgp 19988  df-ur 20005  df-ring 20058  df-rnghom 20251
This theorem is referenced by: (None)
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