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Theorem ecopoveq 6201
 Description: This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation (specified by the hypothesis) in terms of its operation . (Contributed by NM, 16-Aug-1995.)
Hypothesis
Ref Expression
ecopopr.1
Assertion
Ref Expression
ecopoveq
Distinct variable groups:   ,,,,,,   ,,,,,,   ,,,,,,   ,,,,,,   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)

Proof of Theorem ecopoveq
StepHypRef Expression
1 oveq12 5521 . . . 4
2 oveq12 5521 . . . 4
31, 2eqeqan12d 2055 . . 3
43an42s 523 . 2
5 ecopopr.1 . 2
64, 5opbrop 4419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393  cop 3378   class class class wbr 3764  copab 3817   cxp 4343  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  ecopovsym  6202  ecopovtrn  6203  ecopover  6204  ecopovsymg  6205  ecopovtrng  6206  ecopoverg  6207  enqbreq  6454  enrbreq  6819  prsrlem1  6827
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