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Theorem isrim0 18546
 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 18539 . . . . 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 6558 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
43adantl 481 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
5 oveq12 6558 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
65ancoms 468 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
76adantl 481 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
87eleq2d 2673 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RingHom 𝑟) ↔ 𝑓 ∈ (𝑆 RingHom 𝑅)))
94, 8rabeqbidv 3168 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)} = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 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 6577 . . . . . 6 (𝑅 RingHom 𝑆) ∈ V
1514rabex 4740 . . . . 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 6686 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
1817eleq2d 2673 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)}))
19 cnveq 5218 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2672 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RingHom 𝑅) ↔ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2120elrab 3331 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2218, 21syl6bb 275 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ◡ccnv 5037  (class class class)co 6549   ↦ cmpt2 6551   RingHom crh 18535   RingIso crs 18536 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 4709  ax-nul 4717  ax-pr 4833 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 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rngiso 18539 This theorem is referenced by:  isrim  18556  brric2  18568  ringcinv  41824  ringcinvALTV  41848
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