Theorem ecopoveq 7431

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:   ,,,,,,   ,S,,,,,   ,,,,,,   ,,,,,,   ,,,,,,   ,,,,,,

Proof of Theorem ecopoveq

Step	Hyp	Ref	Expression
1		oveq12 6305	. . . 4
2		oveq12 6305	. . . 4
3	1, 2	eqeqan12d 2480	. . 3
4	3	an42s 827	. 2
5		ecopopr.1	. 2
6	4, 5	opbrop 5084	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035   class class class wbr 4452  {copab 4509  X.cxp 5002  (class class class)co 6296

This theorem is referenced by:  ecopovsym  7432  ecopovtrn  7433  ecopover  7434  enqbreq  9318  enrbreq  9462  prsrlem1  9470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-iota 5556  df-fv 5601  df-ov 6299