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Theorem isrngisom 42419
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 42411 . . . . 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 6805 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
43adantl 467 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5 oveq12 6805 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
65ancoms 446 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
76adantl 467 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
87eleq2d 2836 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ 𝑓 ∈ (𝑆 RngHomo 𝑅)))
94, 8rabeqbidv 3345 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
10 elex 3364 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 466 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3364 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 467 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 6827 . . . . . 6 (𝑅 RngHomo 𝑆) ∈ V
1514rabex 4947 . . . . 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 6939 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)})
1817eleq2d 2836 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19 cnveq 5433 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2835 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2120elrab 3515 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ 𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅)))
2218, 21syl6bb 276 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 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  ccnv 5249  (class class class)co 6796  cmpt2 6798   RngHomo crngh 42408   RngIsom crngs 42409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-rngisom 42411
This theorem is referenced by:  isrngim  42427  rngcinv  42504  rngcinvALTV  42516
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