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Theorem isrim0 18925
Description: An isomorphism of rings is a homomorphism whose converse is also a homomorphism . (Contributed by AV, 22-Oct-2019.)
Assertion
Ref Expression
isrim0 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))

Proof of Theorem isrim0
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngiso 18918 . . . . 5 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
21a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)}))
3 oveq12 6822 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
43adantl 473 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
5 oveq12 6822 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
65ancoms 468 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
76adantl 473 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
87eleq2d 2825 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RingHom 𝑟) ↔ 𝑓 ∈ (𝑆 RingHom 𝑅)))
94, 8rabeqbidv 3335 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)} = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
10 elex 3352 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 472 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3352 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 473 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 6841 . . . . . 6 (𝑅 RingHom 𝑆) ∈ V
1514rabex 4964 . . . . 5 {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V
1615a1i 11 . . . 4 ((𝑅𝑉𝑆𝑊) → {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ∈ V)
172, 9, 11, 13, 16ovmpt2d 6953 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
1817eleq2d 2825 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)}))
19 cnveq 5451 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2824 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RingHom 𝑅) ↔ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2120elrab 3504 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2218, 21syl6bb 276 1 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  ccnv 5265  (class class class)co 6813  cmpt2 6815   RingHom crh 18914   RingIso crs 18915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-rngiso 18918
This theorem is referenced by:  isrim  18935  brric2  18947  ringcinv  42542  ringcinvALTV  42566
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