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Theorem rnghmsscmap 46362
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 4193 . . 3 (Rng ∩ 𝑈) ⊆ 𝑈
31, 2eqsstrdi 4002 . 2 (𝜑𝑅𝑈)
4 eqid 2733 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
5 eqid 2733 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
64, 5rnghmf 46287 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
7 simpr 486 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
8 fvex 6859 . . . . . . . . . 10 (Base‘𝑏) ∈ V
9 fvex 6859 . . . . . . . . . 10 (Base‘𝑎) ∈ V
108, 9pm3.2i 472 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11 elmapg 8784 . . . . . . . . 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 414 . . . . . 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 3954 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
17 ovres 7524 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1817adantl 483 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
19 eqidd 2734 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
20 fveq2 6846 . . . . . . 7 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21 fveq2 6846 . . . . . . 7 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2220, 21oveqan12rd 7381 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
2322adantl 483 . . . . 5 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
243sseld 3947 . . . . . . . 8 (𝜑 → (𝑎𝑅𝑎𝑈))
2524com12 32 . . . . . . 7 (𝑎𝑅 → (𝜑𝑎𝑈))
2625adantr 482 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑎𝑈))
2726impcom 409 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑎𝑈)
283sseld 3947 . . . . . . . 8 (𝜑 → (𝑏𝑅𝑏𝑈))
2928com12 32 . . . . . . 7 (𝑏𝑅 → (𝜑𝑏𝑈))
3029adantl 483 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑏𝑈))
3130impcom 409 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑏𝑈)
32 ovexd 7396 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
3319, 23, 27, 31, 32ovmpod 7511 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎)))
3416, 18, 333sstr4d 3995 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
3534ralrimivva 3194 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏))
36 rnghmfn 46278 . . . . 5 RngHomo Fn (Rng × Rng)
3736a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
38 inss1 4192 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
391, 38eqsstrdi 4002 . . . . 5 (𝜑𝑅 ⊆ Rng)
40 xpss12 5652 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
4139, 39, 40syl2anc 585 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
42 fnssres 6628 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
4337, 41, 42syl2anc 585 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
44 eqid 2733 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
45 ovex 7394 . . . . 5 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
4644, 45fnmpoi 8006 . . . 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 3465 . . . 4 (𝑈𝑉𝑈 ∈ V)
5048, 49syl 17 . . 3 (𝜑𝑈 ∈ V)
5143, 47, 50isssc 17711 . 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 712 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  cin 3913  wss 3914   class class class wbr 5109   × cxp 5635  cres 5639   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361  cmpo 7363  m cmap 8771  Basecbs 17091  cat cssc 17698  Rngcrng 46262   RngHomo crngh 46273
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-ixp 8842  df-ssc 17701  df-ghm 19014  df-abl 19573  df-rng 46263  df-rnghomo 46275
This theorem is referenced by:  rnghmsubcsetc  46365
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