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Theorem rngoisoval 34108
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 6805 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
2 fveq2 6333 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3syl6eqr 2823 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5490 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6syl6eqr 2823 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
8 f1oeq2 6270 . . . . 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 6333 . . . . . . . 8 (𝑠 = 𝑆 → (1st𝑠) = (1st𝑆))
11 rngisoval.3 . . . . . . . 8 𝐽 = (1st𝑆)
1210, 11syl6eqr 2823 . . . . . . 7 (𝑠 = 𝑆 → (1st𝑠) = 𝐽)
1312rneqd 5490 . . . . . 6 (𝑠 = 𝑆 → ran (1st𝑠) = ran 𝐽)
14 rngisoval.4 . . . . . 6 𝑌 = ran 𝐽
1513, 14syl6eqr 2823 . . . . 5 (𝑠 = 𝑆 → ran (1st𝑠) = 𝑌)
16 f1oeq3 6271 . . . . 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 499 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
191, 18rabeqbidv 3345 . 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 34107 . 2 RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})
21 ovex 6827 . . 3 (𝑅 RngHom 𝑆) ∈ V
2221rabex 4947 . 2 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌} ∈ V
2319, 20, 22ovmpt2a 6942 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 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  ran crn 5251  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  1st c1st 7317  RingOpscrngo 34025   RngHom crnghom 34091   RngIso crngiso 34092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-rngoiso 34107
This theorem is referenced by:  isrngoiso  34109
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