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Theorem isrim0OLD 20155
Description: Obsolete version of isrim0 20157 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
isrim0OLD ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))

Proof of Theorem isrim0OLD
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngiso 20148 . . . . 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 7367 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
43adantl 483 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑟 RingHom 𝑠) = (𝑅 RingHom 𝑆))
5 oveq12 7367 . . . . . . . 8 ((𝑠 = 𝑆𝑟 = 𝑅) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
65ancoms 460 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
76adantl 483 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑠 RingHom 𝑟) = (𝑆 RingHom 𝑅))
87eleq2d 2824 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (𝑓 ∈ (𝑠 RingHom 𝑟) ↔ 𝑓 ∈ (𝑆 RingHom 𝑅)))
94, 8rabeqbidv 3425 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)} = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
10 elex 3464 . . . . 5 (𝑅𝑉𝑅 ∈ V)
1110adantr 482 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
12 elex 3464 . . . . 5 (𝑆𝑊𝑆 ∈ V)
1312adantl 483 . . . 4 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
14 ovex 7391 . . . . . 6 (𝑅 RingHom 𝑆) ∈ V
1514rabex 5290 . . . . 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, 16ovmpod 7508 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)})
1817eleq2d 2824 . 2 ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)}))
19 cnveq 5830 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2019eleq1d 2823 . . 3 (𝑓 = 𝐹 → (𝑓 ∈ (𝑆 RingHom 𝑅) ↔ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2120elrab 3646 . 2 (𝐹 ∈ {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓 ∈ (𝑆 RingHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅)))
2218, 21bitrdi 287 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 205  wa 397   = wceq 1542  wcel 2107  {crab 3408  Vcvv 3446  ccnv 5633  (class class class)co 7358  cmpo 7360   RingHom crh 20144   RingIso crs 20145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-rngiso 20148
This theorem is referenced by:  isrimOLD  20167
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