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Theorem rnghmsscmap2 42772
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 ssidd 3820 . 2 (𝜑𝑅𝑅)
2 eqid 2799 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
3 eqid 2799 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
42, 3rnghmf 42698 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
5 simpr 478 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
6 fvex 6424 . . . . . . . . . 10 (Base‘𝑏) ∈ V
7 fvex 6424 . . . . . . . . . 10 (Base‘𝑎) ∈ V
86, 7pm3.2i 463 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9 elmapg 8108 . . . . . . . . 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 249 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1211ex 402 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
134, 12syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHomo 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1413ssrdv 3804 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
15 ovres 7034 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1615adantl 474 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
17 eqidd 2800 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
18 fveq2 6411 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19 fveq2 6411 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2018, 19oveqan12rd 6898 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2120adantl 474 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
22 simpl 475 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
23 simpr 478 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
24 ovexd 6912 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
2517, 21, 22, 23, 24ovmpt2d 7022 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2625adantl 474 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2714, 16, 263sstr4d 3844 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
2827ralrimivva 3152 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
29 rnghmfn 42689 . . . . 5 RngHomo Fn (Rng × Rng)
3029a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
31 rnghmsscmap.r . . . . . 6 (𝜑𝑅 = (Rng ∩ 𝑈))
32 inss1 4028 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
3331, 32syl6eqss 3851 . . . . 5 (𝜑𝑅 ⊆ Rng)
34 xpss12 5327 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
3533, 33, 34syl2anc 580 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
36 fnssres 6215 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3730, 35, 36syl2anc 580 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38 eqid 2799 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
39 ovex 6910 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4038, 39fnmpt2i 7475 . . . 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 4003 . . . . 5 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
43 rnghmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
44 inex1g 4996 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4543, 44syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Rng) ∈ V)
4642, 45syl5eqel 2882 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
4731, 46eqeltrd 2878 . . 3 (𝜑𝑅 ∈ V)
4837, 41, 47isssc 16794 . 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))))
491, 28, 48mpbir2and 705 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  cin 3768  wss 3769   class class class wbr 4843   × cxp 5310  cres 5314   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  cmpt2 6880  𝑚 cmap 8095  Basecbs 16184  cat cssc 16781  Rngcrng 42673   RngHomo crngh 42684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097  df-ixp 8149  df-ssc 16784  df-ghm 17971  df-abl 18511  df-rng0 42674  df-rnghomo 42686
This theorem is referenced by: (None)
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