Theorem opelcoOLD 4131

Description: Ordered pair membership in a composition.

Hypotheses

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

Assertion

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

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

Proof of Theorem opelcoOLD

Step	Hyp	Ref	Expression
1		df-co 4003	. . 3 |- (C o. D) = {<.y, z>. | E.x(yDx /\ xCz)}
2	1	eleq2i 1961	. 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 3341	. . . . 5 |- (y = A -> (yDx <-> ADx))
6	5	anbi1d 679	. . . 4 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
7	6	exbidv 1657	. . 3 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
8		breq2 3342	. . . . 5 |- (z = B -> (xCz <-> xCB))
9	8	anbi2d 678	. . . 4 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
10	9	exbidv 1657	. . 3 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
11	3, 4, 7, 10	opelopab 3570	. 2 |- (<.A, B>. e. {<.y, z>. | E.x(yDx /\ xCz)} <-> E.x(ADx /\ xCB))
12	2, 11	bitri 190	1 |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292  <.cop 3046   class class class wbr 3338  {copab 3395   o. ccom 3990

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-co 4003

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