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Theorem rnghmsscmap2 41738
Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rnghmsscmap.u (𝜑𝑈𝑉)
rnghmsscmap.r (𝜑𝑅 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rnghmsscmap2 (𝜑 → ( RngHomo ↾ (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 rnghmsscmap2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3616 . . 3 𝑅𝑅
21a1i 11 . 2 (𝜑𝑅𝑅)
3 eqid 2620 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
4 eqid 2620 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
53, 4rnghmf 41664 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
6 simpr 477 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
7 fvex 6188 . . . . . . . . . 10 (Base‘𝑏) ∈ V
8 fvex 6188 . . . . . . . . . 10 (Base‘𝑎) ∈ V
97, 8pm3.2i 471 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
10 elmapg 7855 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
119, 10mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
126, 11mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1312ex 450 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
145, 13syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHomo 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1514ssrdv 3601 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
16 ovres 6785 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1716adantl 482 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
18 eqidd 2621 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
19 fveq2 6178 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
20 fveq2 6178 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2119, 20oveqan12rd 6655 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2221adantl 482 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
23 simpl 473 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
24 simpr 477 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
25 ovexd 6665 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
2618, 22, 23, 24, 25ovmpt2d 6773 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2726adantl 482 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2815, 17, 273sstr4d 3640 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
2928ralrimivva 2968 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
30 rnghmfn 41655 . . . . 5 RngHomo Fn (Rng × Rng)
3130a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
32 rnghmsscmap.r . . . . . 6 (𝜑𝑅 = (Rng ∩ 𝑈))
33 inss1 3825 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
3432, 33syl6eqss 3647 . . . . 5 (𝜑𝑅 ⊆ Rng)
35 xpss12 5215 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
3634, 34, 35syl2anc 692 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
37 fnssres 5992 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3831, 36, 37syl2anc 692 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
39 eqid 2620 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
40 ovex 6663 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4139, 40fnmpt2i 7224 . . . 4 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅)
4241a1i 11 . . 3 (𝜑 → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅))
43 incom 3797 . . . . 5 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
44 rnghmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
45 inex1g 4792 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4644, 45syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Rng) ∈ V)
4743, 46syl5eqel 2703 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
4832, 47eqeltrd 2699 . . 3 (𝜑𝑅 ∈ V)
4938, 42, 48isssc 16461 . 2 (𝜑 → (( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑅 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
502, 29, 49mpbir2and 956 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cin 3566  wss 3567   class class class wbr 4644   × cxp 5102  cres 5106   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑚 cmap 7842  Basecbs 15838  cat cssc 16448  Rngcrng 41639   RngHomo crngh 41650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-ixp 7894  df-ssc 16451  df-ghm 17639  df-abl 18177  df-rng0 41640  df-rnghomo 41652
This theorem is referenced by: (None)
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