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Theorem isrngoiso 28931
 Description: The predicate "is a ring isomorphism between and ." (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngisoval.1
rngisoval.2
rngisoval.3
rngisoval.4
Assertion
Ref Expression
isrngoiso

Proof of Theorem isrngoiso
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rngisoval.1 . . . 4
2 rngisoval.2 . . . 4
3 rngisoval.3 . . . 4
4 rngisoval.4 . . . 4
51, 2, 3, 4rngoisoval 28930 . . 3
65eleq2d 2524 . 2
7 f1oeq1 5739 . . 3
87elrab 3222 . 2
96, 8syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758  crab 2802   crn 4948  wf1o 5524  cfv 5525  (class class class)co 6199  c1st 6684  crngo 24013   crnghom 28913   crngiso 28914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-rngoiso 28929 This theorem is referenced by:  rngoiso1o  28932  rngoisohom  28933  rngoisocnv  28934  rngoisoco  28935
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