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Theorem rngoisoval 35136
Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngisoval.1 𝐺 = (1st𝑅)
rngisoval.2 𝑋 = ran 𝐺
rngisoval.3 𝐽 = (1st𝑆)
rngisoval.4 𝑌 = ran 𝐽
Assertion
Ref Expression
rngoisoval ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓   𝑓,𝑋   𝑓,𝑌
Allowed substitution hints:   𝐺(𝑓)   𝐽(𝑓)

Proof of Theorem rngoisoval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7154 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
2 fveq2 6663 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3syl6eqr 2871 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5801 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6syl6eqr 2871 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87f1oeq2d 6604 . . . 4 (𝑟 = 𝑅 → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto→ran (1st𝑠)))
9 fveq2 6663 . . . . . . . 8 (𝑠 = 𝑆 → (1st𝑠) = (1st𝑆))
10 rngisoval.3 . . . . . . . 8 𝐽 = (1st𝑆)
119, 10syl6eqr 2871 . . . . . . 7 (𝑠 = 𝑆 → (1st𝑠) = 𝐽)
1211rneqd 5801 . . . . . 6 (𝑠 = 𝑆 → ran (1st𝑠) = ran 𝐽)
13 rngisoval.4 . . . . . 6 𝑌 = ran 𝐽
1412, 13syl6eqr 2871 . . . . 5 (𝑠 = 𝑆 → ran (1st𝑠) = 𝑌)
1514f1oeq3d 6605 . . . 4 (𝑠 = 𝑆 → (𝑓:𝑋1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
168, 15sylan9bb 510 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
171, 16rabeqbidv 3483 . 2 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
18 df-rngoiso 35135 . 2 RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})
19 ovex 7178 . . 3 (𝑅 RngHom 𝑆) ∈ V
2019rabex 5226 . 2 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌} ∈ V
2117, 18, 20ovmpoa 7294 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  ran crn 5549  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  1st c1st 7676  RingOpscrngo 35053   RngHom crnghom 35119   RngIso crngiso 35120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rngoiso 35135
This theorem is referenced by:  isrngoiso  35137
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