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Theorem isrngisom 41678
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 41670 . . . . 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 6536 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
43adantl 481 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5 oveq12 6536 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
65ancoms 468 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
76adantl 481 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
87eleq2d 2673 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ 𝑓 ∈ (𝑆 RngHomo 𝑅)))
94, 8rabeqbidv 3168 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
10 elex 3185 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 480 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3185 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 481 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 6555 . . . . . 6 (𝑅 RngHomo 𝑆) ∈ V
1514rabex 4735 . . . . 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 6664 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
1817eleq2d 2673 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19 cnveq 5206 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2672 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2120elrab 3331 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2218, 21syl6bb 275 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 195  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  ccnv 5027  (class class class)co 6527  cmpt2 6529   RngHomo crngh 41667   RngIsom crngs 41668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-rngisom 41670
This theorem is referenced by:  isrngim  41686  rngcinv  41765  rngcinvALTV  41777
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