Theorem opelco 2509

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 group(s):   x,A   x,B   x,C   x,D

Proof of Theorem opelco

Step	Hyp	Ref	Expression
1		df-co 2427	. . 3 |- (C o. D) = {<.y, z>. | E.x(yDx /\ xCz)}
2	1	eleq2i 1153	. 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 2065	. . . . 5 |- (y = A -> (yDx <-> ADx))
6	5	anbi1d 469	. . . 4 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
7	6	biexdv 936	. . 3 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
8		breq2 2066	. . . . 5 |- (z = B -> (xCz <-> xCB))
9	8	anbi2d 468	. . . 4 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
10	9	biexdv 936	. . 3 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
11	3, 4, 7, 10	opelopab 2117	. 2 |- (<.A, B>. e. {<.y, z>. | E.x(yDx /\ xCz)} <-> E.x(ADx /\ xCB))
12	2, 11	bitr 151	1 |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   class class class wbr 2054  {copab 2055   o. ccom 2414

This theorem is referenced by:  brco 2510  opelcog 2511  cnvco 2520  dmco 2570  dmcosseq 2572  cores 2659  co02 2663  coi1 2665  coass 2667  dmco2 2673

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-co 2427

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