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Theorem rngoisoval 26593
 Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngisoval.1
rngisoval.2
rngisoval.3
rngisoval.4
Assertion
Ref Expression
rngoisoval
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem rngoisoval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6090 . . 3
2 fveq2 5728 . . . . . . . 8
3 rngisoval.1 . . . . . . . 8
42, 3syl6eqr 2486 . . . . . . 7
54rneqd 5097 . . . . . 6
6 rngisoval.2 . . . . . 6
75, 6syl6eqr 2486 . . . . 5
8 f1oeq2 5666 . . . . 5
97, 8syl 16 . . . 4
10 fveq2 5728 . . . . . . . 8
11 rngisoval.3 . . . . . . . 8
1210, 11syl6eqr 2486 . . . . . . 7
1312rneqd 5097 . . . . . 6
14 rngisoval.4 . . . . . 6
1513, 14syl6eqr 2486 . . . . 5
16 f1oeq3 5667 . . . . 5
1715, 16syl 16 . . . 4
189, 17sylan9bb 681 . . 3
191, 18rabeqbidv 2951 . 2
20 df-rngoiso 26592 . 2
21 ovex 6106 . . 3
2221rabex 4354 . 2
2319, 20, 22ovmpt2a 6204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2709   crn 4879  wf1o 5453  cfv 5454  (class class class)co 6081  c1st 6347  crngo 21963   crnghom 26576   crngiso 26577 This theorem is referenced by:  isrngoiso  26594 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-rngoiso 26592
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