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Theorem rngoisoval 35414
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 7148 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
2 fveq2 6649 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3eqtr4di 2854 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5776 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2854 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87f1oeq2d 6590 . . . 4 (𝑟 = 𝑅 → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto→ran (1st𝑠)))
9 fveq2 6649 . . . . . . . 8 (𝑠 = 𝑆 → (1st𝑠) = (1st𝑆))
10 rngisoval.3 . . . . . . . 8 𝐽 = (1st𝑆)
119, 10eqtr4di 2854 . . . . . . 7 (𝑠 = 𝑆 → (1st𝑠) = 𝐽)
1211rneqd 5776 . . . . . 6 (𝑠 = 𝑆 → ran (1st𝑠) = ran 𝐽)
13 rngisoval.4 . . . . . 6 𝑌 = ran 𝐽
1412, 13eqtr4di 2854 . . . . 5 (𝑠 = 𝑆 → ran (1st𝑠) = 𝑌)
1514f1oeq3d 6591 . . . 4 (𝑠 = 𝑆 → (𝑓:𝑋1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
168, 15sylan9bb 513 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠) ↔ 𝑓:𝑋1-1-onto𝑌))
171, 16rabeqbidv 3436 . 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 35413 . 2 RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})
19 ovex 7172 . . 3 (𝑅 RngHom 𝑆) ∈ V
2019rabex 5202 . 2 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌} ∈ V
2117, 18, 20ovmpoa 7288 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 399   = wceq 1538  wcel 2112  {crab 3113  ran crn 5524  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  1st c1st 7673  RingOpscrngo 35331   RngHom crnghom 35397   RngIso crngiso 35398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rngoiso 35413
This theorem is referenced by:  isrngoiso  35415
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