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Theorem rhmsscrnghm 47009
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
Hypotheses
Ref Expression
rhmsscrnghm.u (𝜑𝑈𝑉)
rhmsscrnghm.r (𝜑𝑅 = (Ring ∩ 𝑈))
rhmsscrnghm.s (𝜑𝑆 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rhmsscrnghm (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))

Proof of Theorem rhmsscrnghm
Dummy variables 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringrng 46740 . . . . . 6 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
21a1i 11 . . . . 5 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
32ssrdv 3988 . . . 4 (𝜑 → Ring ⊆ Rng)
43ssrind 4235 . . 3 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
5 rhmsscrnghm.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
6 rhmsscrnghm.s . . 3 (𝜑𝑆 = (Rng ∩ 𝑈))
74, 5, 63sstr4d 4029 . 2 (𝜑𝑅𝑆)
8 ovres 7575 . . . . . . 7 ((𝑥𝑅𝑦𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
98adantl 482 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
109eleq2d 2819 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ∈ (𝑥 RingHom 𝑦)))
11 rhmisrnghm 46808 . . . . . 6 ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥 RngHomo 𝑦))
127sseld 3981 . . . . . . . . . 10 (𝜑 → (𝑥𝑅𝑥𝑆))
137sseld 3981 . . . . . . . . . 10 (𝜑 → (𝑦𝑅𝑦𝑆))
1412, 13anim12d 609 . . . . . . . . 9 (𝜑 → ((𝑥𝑅𝑦𝑅) → (𝑥𝑆𝑦𝑆)))
1514imp 407 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥𝑆𝑦𝑆))
16 ovres 7575 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1715, 16syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1817eleq2d 2819 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ∈ (𝑥 RngHomo 𝑦)))
1911, 18imbitrrid 245 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2010, 19sylbid 239 . . . 4 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2120ssrdv 3988 . . 3 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
2221ralrimivva 3200 . 2 (𝜑 → ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
23 inss1 4228 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
245, 23eqsstrdi 4036 . . . . 5 (𝜑𝑅 ⊆ Ring)
25 xpss12 5691 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
2624, 24, 25syl2anc 584 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27 rhmfn 46806 . . . . 5 RingHom Fn (Ring × Ring)
28 fnssresb 6672 . . . . 5 ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
2927, 28mp1i 13 . . . 4 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
3026, 29mpbird 256 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31 inss1 4228 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
326, 31eqsstrdi 4036 . . . . 5 (𝜑𝑆 ⊆ Rng)
33 xpss12 5691 . . . . 5 ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
3432, 32, 33syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35 rnghmfn 46773 . . . . 5 RngHomo Fn (Rng × Rng)
36 fnssresb 6672 . . . . 5 ( RngHomo Fn (Rng × Rng) → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3735, 36mp1i 13 . . . 4 (𝜑 → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3834, 37mpbird 256 . . 3 (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39 rhmsscrnghm.u . . . . 5 (𝜑𝑈𝑉)
40 incom 4201 . . . . . 6 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41 inex1g 5319 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4240, 41eqeltrid 2837 . . . . 5 (𝑈𝑉 → (Rng ∩ 𝑈) ∈ V)
4339, 42syl 17 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
446, 43eqeltrd 2833 . . 3 (𝜑𝑆 ∈ V)
4530, 38, 44isssc 17769 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)) ↔ (𝑅𝑆 ∧ ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))))
467, 22, 45mpbir2and 711 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  cin 3947  wss 3948   class class class wbr 5148   × cxp 5674  cres 5678   Fn wfn 6538  (class class class)co 7411  cat cssc 17756  Ringcrg 20058   RingHom crh 20252  Rngcrng 46733   RngHomo crngh 46768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-plusg 17212  df-0g 17389  df-ssc 17759  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-grp 18824  df-minusg 18825  df-ghm 19092  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-ring 20060  df-rnghom 20255  df-mgmhm 46634  df-rng 46734  df-rnghomo 46770
This theorem is referenced by:  rhmsubcrngc  47012  rhmsubc  47073  rhmsubcALTV  47091
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