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Theorem rngoisoval 36439
Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngisoval.1 𝐺 = (1st β€˜π‘…)
rngisoval.2 𝑋 = ran 𝐺
rngisoval.3 𝐽 = (1st β€˜π‘†)
rngisoval.4 π‘Œ = ran 𝐽
Assertion
Ref Expression
rngoisoval ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ})
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓   𝑓,𝑋   𝑓,π‘Œ
Allowed substitution hints:   𝐺(𝑓)   𝐽(𝑓)

Proof of Theorem rngoisoval
Dummy variables π‘Ÿ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7367 . . 3 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (π‘Ÿ RngHom 𝑠) = (𝑅 RngHom 𝑆))
2 fveq2 6843 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
42, 3eqtr4di 2795 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
54rneqd 5894 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2795 . . . . 5 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = 𝑋)
87f1oeq2d 6781 . . . 4 (π‘Ÿ = 𝑅 β†’ (𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ ) ↔ 𝑓:𝑋–1-1-ontoβ†’ran (1st β€˜π‘ )))
9 fveq2 6843 . . . . . . . 8 (𝑠 = 𝑆 β†’ (1st β€˜π‘ ) = (1st β€˜π‘†))
10 rngisoval.3 . . . . . . . 8 𝐽 = (1st β€˜π‘†)
119, 10eqtr4di 2795 . . . . . . 7 (𝑠 = 𝑆 β†’ (1st β€˜π‘ ) = 𝐽)
1211rneqd 5894 . . . . . 6 (𝑠 = 𝑆 β†’ ran (1st β€˜π‘ ) = ran 𝐽)
13 rngisoval.4 . . . . . 6 π‘Œ = ran 𝐽
1412, 13eqtr4di 2795 . . . . 5 (𝑠 = 𝑆 β†’ ran (1st β€˜π‘ ) = π‘Œ)
1514f1oeq3d 6782 . . . 4 (𝑠 = 𝑆 β†’ (𝑓:𝑋–1-1-ontoβ†’ran (1st β€˜π‘ ) ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
168, 15sylan9bb 511 . . 3 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ ) ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
171, 16rabeqbidv 3425 . 2 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ {𝑓 ∈ (π‘Ÿ RngHom 𝑠) ∣ 𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ )} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ})
18 df-rngoiso 36438 . 2 RngIso = (π‘Ÿ ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (π‘Ÿ RngHom 𝑠) ∣ 𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ )})
19 ovex 7391 . . 3 (𝑅 RngHom 𝑆) ∈ V
2019rabex 5290 . 2 {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ} ∈ V
2117, 18, 20ovmpoa 7511 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3408  ran crn 5635  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  RingOpscrngo 36356   RngHom crnghom 36422   RngIso crngiso 36423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-rngoiso 36438
This theorem is referenced by:  isrngoiso  36440
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