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Theorem rnghmsscmap 42484
Description: The non-unital ring homomorphisms between non-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
rnghmsscmap.u (𝜑𝑈𝑉)
rnghmsscmap.r (𝜑𝑅 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rnghmsscmap (𝜑 → ( RngHomo ↾ (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 rnghmsscmap
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghmsscmap.r . . 3 (𝜑𝑅 = (Rng ∩ 𝑈))
2 inss2 3977 . . 3 (Rng ∩ 𝑈) ⊆ 𝑈
31, 2syl6eqss 3796 . 2 (𝜑𝑅𝑈)
4 eqid 2760 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
5 eqid 2760 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
64, 5rnghmf 42409 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
7 simpr 479 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
8 fvex 6362 . . . . . . . . . 10 (Base‘𝑏) ∈ V
9 fvex 6362 . . . . . . . . . 10 (Base‘𝑎) ∈ V
108, 9pm3.2i 470 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11 elmapg 8036 . . . . . . . . 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 247 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1413ex 449 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
156, 14syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHomo 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1615ssrdv 3750 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
17 ovres 6965 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1817adantl 473 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
19 eqidd 2761 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
20 fveq2 6352 . . . . . . 7 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21 fveq2 6352 . . . . . . 7 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2220, 21oveqan12rd 6833 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2322adantl 473 . . . . 5 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
243sseld 3743 . . . . . . . 8 (𝜑 → (𝑎𝑅𝑎𝑈))
2524com12 32 . . . . . . 7 (𝑎𝑅 → (𝜑𝑎𝑈))
2625adantr 472 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑎𝑈))
2726impcom 445 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑎𝑈)
283sseld 3743 . . . . . . . 8 (𝜑 → (𝑏𝑅𝑏𝑈))
2928com12 32 . . . . . . 7 (𝑏𝑅 → (𝜑𝑏𝑈))
3029adantl 473 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑏𝑈))
3130impcom 445 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑏𝑈)
32 ovexd 6843 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
3319, 23, 27, 31, 32ovmpt2d 6953 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
3416, 18, 333sstr4d 3789 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
3534ralrimivva 3109 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
36 rnghmfn 42400 . . . . 5 RngHomo Fn (Rng × Rng)
3736a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
38 inss1 3976 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
391, 38syl6eqss 3796 . . . . 5 (𝜑𝑅 ⊆ Rng)
40 xpss12 5281 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
4139, 39, 40syl2anc 696 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42 fnssres 6165 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
4337, 41, 42syl2anc 696 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44 eqid 2760 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
45 ovex 6841 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4644, 45fnmpt2i 7407 . . . 4 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈)
4746a1i 11 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈))
48 rnghmsscmap.u . . . 4 (𝜑𝑈𝑉)
49 elex 3352 . . . 4 (𝑈𝑉𝑈 ∈ V)
5048, 49syl 17 . . 3 (𝜑𝑈 ∈ V)
5143, 47, 50isssc 16681 . 2 (𝜑 → (( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑈 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
523, 35, 51mpbir2and 995 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cin 3714  wss 3715   class class class wbr 4804   × cxp 5264  cres 5268   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Basecbs 16059  cat cssc 16668  Rngcrng 42384   RngHomo crngh 42395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-ixp 8075  df-ssc 16671  df-ghm 17859  df-abl 18396  df-rng0 42385  df-rnghomo 42397
This theorem is referenced by:  rnghmsubcsetc  42487
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