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Theorem rngoisoval 37358
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 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 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 7414 . . 3 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (π‘Ÿ RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2 fveq2 6885 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
3 rngisoval.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
42, 3eqtr4di 2784 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
54rneqd 5931 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = ran 𝐺)
6 rngisoval.2 . . . . . 6 𝑋 = ran 𝐺
75, 6eqtr4di 2784 . . . . 5 (π‘Ÿ = 𝑅 β†’ ran (1st β€˜π‘Ÿ) = 𝑋)
87f1oeq2d 6823 . . . 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 6885 . . . . . . . 8 (𝑠 = 𝑆 β†’ (1st β€˜π‘ ) = (1st β€˜π‘†))
10 rngisoval.3 . . . . . . . 8 𝐽 = (1st β€˜π‘†)
119, 10eqtr4di 2784 . . . . . . 7 (𝑠 = 𝑆 β†’ (1st β€˜π‘ ) = 𝐽)
1211rneqd 5931 . . . . . 6 (𝑠 = 𝑆 β†’ ran (1st β€˜π‘ ) = ran 𝐽)
13 rngisoval.4 . . . . . 6 π‘Œ = ran 𝐽
1412, 13eqtr4di 2784 . . . . 5 (𝑠 = 𝑆 β†’ ran (1st β€˜π‘ ) = π‘Œ)
1514f1oeq3d 6824 . . . 4 (𝑠 = 𝑆 β†’ (𝑓:𝑋–1-1-ontoβ†’ran (1st β€˜π‘ ) ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
168, 15sylan9bb 509 . . 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 3443 . 2 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ {𝑓 ∈ (π‘Ÿ RingOpsHom 𝑠) ∣ 𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ )} = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ})
18 df-rngoiso 37357 . 2 RingOpsIso = (π‘Ÿ ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (π‘Ÿ RingOpsHom 𝑠) ∣ 𝑓:ran (1st β€˜π‘Ÿ)–1-1-ontoβ†’ran (1st β€˜π‘ )})
19 ovex 7438 . . 3 (𝑅 RingOpsHom 𝑆) ∈ V
2019rabex 5325 . 2 {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ} ∈ V
2117, 18, 20ovmpoa 7559 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-ontoβ†’π‘Œ})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  ran crn 5670  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  RingOpscrngo 37275   RingOpsHom crngohom 37341   RingOpsIso crngoiso 37342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rngoiso 37357
This theorem is referenced by:  isrngoiso  37359
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