Theorem opelco 3288

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 3187	. . 3 |- (C o. D) = {<.y, z>. | E.x(yDx /\ xCz)}
2	1	eleq2i 1538	. 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 2622	. . . . 5 |- (y = A -> (yDx <-> ADx))
6	5	anbi1d 617	. . . 4 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
7	6	exbidv 1279	. . 3 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
8		breq2 2623	. . . . 5 |- (z = B -> (xCz <-> xCB))
9	8	anbi2d 616	. . . 4 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
10	9	exbidv 1279	. . 3 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
11	3, 4, 7, 10	opelopab 2820	. 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 956   e. wcel 958  E.wex 980  Vcvv 1811  <.cop 2411   class class class wbr 2619  {copab 2666   o. ccom 3174

This theorem is referenced by:  brco 3289  opelcog 3290  cnvco 3300  dmcoss 3363  dmcosseq 3365  cores 3499  co02 3508  coi1 3510  coass 3512

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-co 3187

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