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Theorem rngoisoval 32946
 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 6558 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
2 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5274 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6syl6eqr 2662 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
8 f1oeq2 6041 . . . . 5 (ran (1st𝑟) = 𝑋 → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto→ran (1st𝑠)))
97, 8syl 17 . . . 4 (𝑟 = 𝑅 → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto→ran (1st𝑠)))
10 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑆 → (1st𝑠) = (1st𝑆))
11 rngisoval.3 . . . . . . . 8 𝐽 = (1st𝑆)
1210, 11syl6eqr 2662 . . . . . . 7 (𝑠 = 𝑆 → (1st𝑠) = 𝐽)
1312rneqd 5274 . . . . . 6 (𝑠 = 𝑆 → ran (1st𝑠) = ran 𝐽)
14 rngisoval.4 . . . . . 6 𝑌 = ran 𝐽
1513, 14syl6eqr 2662 . . . . 5 (𝑠 = 𝑆 → ran (1st𝑠) = 𝑌)
16 f1oeq3 6042 . . . . 5 (ran (1st𝑠) = 𝑌 → (𝑓:𝑋1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
1715, 16syl 17 . . . 4 (𝑠 = 𝑆 → (𝑓:𝑋1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
189, 17sylan9bb 732 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
191, 18rabeqbidv 3168 . 2 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
20 df-rngoiso 32945 . 2 RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})
21 ovex 6577 . . 3 (𝑅 RngHom 𝑆) ∈ V
2221rabex 4740 . 2 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌} ∈ V
2319, 20, 22ovmpt2a 6689 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   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  RingOpscrngo 32863   RngHom crnghom 32929   RngIso crngiso 32930 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-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rngoiso 32945 This theorem is referenced by:  isrngoiso  32947
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