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Theorem isrngisom 42565
Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
Assertion
Ref Expression
isrngisom ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅))))

Proof of Theorem isrngisom
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngisom 42557 . . . . 5 RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)})
21a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)}))
3 oveq12 6851 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
43adantl 473 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5 oveq12 6851 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
65ancoms 450 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
76adantl 473 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
87eleq2d 2830 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ 𝑓 ∈ (𝑆 RngHomo 𝑅)))
94, 8rabeqbidv 3344 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
10 elex 3365 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 472 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3365 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 473 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 6874 . . . . . 6 (𝑅 RngHomo 𝑆) ∈ V
1514rabex 4973 . . . . 5 {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V
1615a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V)
172, 9, 11, 13, 16ovmpt2d 6986 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
1817eleq2d 2830 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19 cnveq 5464 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2829 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2120elrab 3519 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2218, 21syl6bb 278 1 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  ccnv 5276  (class class class)co 6842  cmpt2 6844   RngHomo crngh 42554   RngIsom crngs 42555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-rngisom 42557
This theorem is referenced by:  isrngim  42573  rngcinv  42650  rngcinvALTV  42662
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