Theorem opelco 3277

Description: Ordered pair membership in a composition.

Hypotheses

Ref	Expression
opelco.1	|- A e. V
opelco.2	|- B e. V

Assertion

Ref	Expression
opelco	|- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))

Distinct variable groups:   x,A   x,B   x,C   x,D

Proof of Theorem opelco

Step	Hyp	Ref	Expression
1		df-co 3177	. . 3 |- (C o. D) = {<.y, z>. | E.x(yDx /\ xCz)}
2	1	eleq2i 1530	. 2 |- (<.A, B>. e. (C o. D) <-> <.A, B>. e. {<.y, z>. | E.x(yDx /\ xCz)})
3		opelco.1	. . 3 |- A e. V
4		opelco.2	. . 3 |- B e. V
5		breq1 2612	. . . . 5 |- (y = A -> (yDx <-> ADx))
6	5	anbi1d 615	. . . 4 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
7	6	exbidv 1274	. . 3 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
8		breq2 2613	. . . . 5 |- (z = B -> (xCz <-> xCB))
9	8	anbi2d 614	. . . 4 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
10	9	exbidv 1274	. . 3 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
11	3, 4, 7, 10	opelopab 2809	. 2 |- (<.A, B>. e. {<.y, z>. | E.x(yDx /\ xCz)} <-> E.x(ADx /\ xCB))
12	2, 11	bitr 173	1 |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  <.cop 2401   class class class wbr 2609  {copab 2656   o. ccom 3164

This theorem is referenced by:  brco 3278  opelcog 3279  cnvco 3289  dmcoss 3347  dmcosseq 3349  cores 3485  co02 3494  coi1 3496  coass 3498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-co 3177

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