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Theorem rngoisoval 37985
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 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 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 7441 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2 fveq2 6905 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5948 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2794 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87f1oeq2d 6843 . . . 4 (𝑟 = 𝑅 → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto→ran (1st𝑠)))
9 fveq2 6905 . . . . . . . 8 (𝑠 = 𝑆 → (1st𝑠) = (1st𝑆))
10 rngisoval.3 . . . . . . . 8 𝐽 = (1st𝑆)
119, 10eqtr4di 2794 . . . . . . 7 (𝑠 = 𝑆 → (1st𝑠) = 𝐽)
1211rneqd 5948 . . . . . 6 (𝑠 = 𝑆 → ran (1st𝑠) = ran 𝐽)
13 rngisoval.4 . . . . . 6 𝑌 = ran 𝐽
1412, 13eqtr4di 2794 . . . . 5 (𝑠 = 𝑆 → ran (1st𝑠) = 𝑌)
1514f1oeq3d 6844 . . . 4 (𝑠 = 𝑆 → (𝑓:𝑋1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
168, 15sylan9bb 509 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
171, 16rabeqbidv 3454 . 2 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)} = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
18 df-rngoiso 37984 . 2 RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})
19 ovex 7465 . . 3 (𝑅 RingOpsHom 𝑆) ∈ V
2019rabex 5338 . 2 {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌} ∈ V
2117, 18, 20ovmpoa 7589 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  ran crn 5685  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  1st c1st 8013  RingOpscrngo 37902   RingOpsHom crngohom 37968   RingOpsIso crngoiso 37969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-rngoiso 37984
This theorem is referenced by:  isrngoiso  37986
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