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Theorem rnghmsscmap 20556
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 → ( RngHom ↾ (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 4225 . . 3 (Rng ∩ 𝑈) ⊆ 𝑈
31, 2eqsstrdi 4032 . 2 (𝜑𝑅𝑈)
4 eqid 2727 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
5 eqid 2727 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
64, 5rnghmf 20380 . . . . . 6 ( ∈ (𝑎 RngHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
7 simpr 484 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
8 fvex 6904 . . . . . . . . . 10 (Base‘𝑏) ∈ V
9 fvex 6904 . . . . . . . . . 10 (Base‘𝑎) ∈ V
108, 9pm3.2i 470 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11 elmapg 8851 . . . . . . . . 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 257 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))
1413ex 412 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
156, 14syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHom 𝑏) → ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
1615ssrdv 3984 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17 ovres 7581 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
1817adantl 481 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏))
19 eqidd 2728 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
20 fveq2 6891 . . . . . . 7 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21 fveq2 6891 . . . . . . 7 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2220, 21oveqan12rd 7434 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2322adantl 481 . . . . 5 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
243sseld 3977 . . . . . . . 8 (𝜑 → (𝑎𝑅𝑎𝑈))
2524com12 32 . . . . . . 7 (𝑎𝑅 → (𝜑𝑎𝑈))
2625adantr 480 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑎𝑈))
2726impcom 407 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑎𝑈)
283sseld 3977 . . . . . . . 8 (𝜑 → (𝑏𝑅𝑏𝑈))
2928com12 32 . . . . . . 7 (𝑏𝑅 → (𝜑𝑏𝑈))
3029adantl 481 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑏𝑈))
3130impcom 407 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑏𝑈)
32 ovexd 7449 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
3319, 23, 27, 31, 32ovmpod 7567 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
3416, 18, 333sstr4d 4025 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
3534ralrimivva 3195 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36 rnghmfn 20371 . . . . 5 RngHom Fn (Rng × Rng)
3736a1i 11 . . . 4 (𝜑 → RngHom Fn (Rng × Rng))
38 inss1 4224 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
391, 38eqsstrdi 4032 . . . . 5 (𝜑𝑅 ⊆ Rng)
40 xpss12 5687 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
4139, 39, 40syl2anc 583 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42 fnssres 6672 . . . 4 (( RngHom Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
4337, 41, 42syl2anc 583 . . 3 (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44 eqid 2727 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45 ovex 7447 . . . . 5 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
4644, 45fnmpoi 8068 . . . 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 3488 . . . 4 (𝑈𝑉𝑈 ∈ V)
5048, 49syl 17 . . 3 (𝜑𝑈 ∈ V)
5143, 47, 50isssc 17796 . 2 (𝜑 → (( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅𝑈 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))))
523, 35, 51mpbir2and 712 1 (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3056  Vcvv 3469  cin 3943  wss 3944   class class class wbr 5142   × cxp 5670  cres 5674   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7414  cmpo 7416  m cmap 8838  Basecbs 17173  cat cssc 17783  Rngcrng 20085   RngHom crnghm 20366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8840  df-ixp 8910  df-ssc 17786  df-ghm 19161  df-abl 19731  df-rng 20086  df-rnghm 20368
This theorem is referenced by:  rnghmsubcsetc  20559
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