Theorem opelcog 3279

Description: Ordered pair membership in a composition.

Assertion

Ref	Expression
opelcog	|- ((A e. R /\ B e. S) -> (<.A, B>. e. (C o. D) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C)))

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

Proof of Theorem opelcog

Step	Hyp	Ref	Expression
1		opeq1 2478	. . . . 5 |- (y = A -> <.y, z>. = <.A, z>.)
2	1	eleq1d 1532	. . . 4 |- (y = A -> (<.y, z>. e. (C o. D) <-> <.A, z>. e. (C o. D)))
3		breq1 2612	. . . . . 6 |- (y = A -> (yDx <-> ADx))
4	3	anbi1d 615	. . . . 5 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
5	4	exbidv 1274	. . . 4 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
6	2, 5	bibi12d 627	. . 3 |- (y = A -> ((<.y, z>. e. (C o. D) <-> E.x(yDx /\ xCz)) <-> (<.A, z>. e. (C o. D) <-> E.x(ADx /\ xCz))))
7		opeq2 2479	. . . . 5 |- (z = B -> <.A, z>. = <.A, B>.)
8	7	eleq1d 1532	. . . 4 |- (z = B -> (<.A, z>. e. (C o. D) <-> <.A, B>. e. (C o. D)))
9		breq2 2613	. . . . . 6 |- (z = B -> (xCz <-> xCB))
10	9	anbi2d 614	. . . . 5 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
11	10	exbidv 1274	. . . 4 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
12	8, 11	bibi12d 627	. . 3 |- (z = B -> ((<.A, z>. e. (C o. D) <-> E.x(ADx /\ xCz)) <-> (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))))
13		visset 1804	. . . 4 |- y e. V
14		visset 1804	. . . 4 |- z e. V
15	13, 14	opelco 3277	. . 3 |- (<.y, z>. e. (C o. D) <-> E.x(yDx /\ xCz))
16	6, 12, 15	vtocl2g 1841	. 2 |- ((A e. R /\ B e. S) -> (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB)))
17		df-br 2610	. . . 4 |- (ADx <-> <.A, x>. e. D)
18		df-br 2610	. . . 4 |- (xCB <-> <.x, B>. e. C)
19	17, 18	anbi12i 481	. . 3 |- ((ADx /\ xCB) <-> (<.A, x>. e. D /\ <.x, B>. e. C))
20	19	exbii 1047	. 2 |- (E.x(ADx /\ xCB) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C))
21	16, 20	syl6bb 534	1 |- ((A e. R /\ B e. S) -> (<.A, B>. e. (C o. D) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  <.cop 2401   class class class wbr 2609   o. ccom 3164

This theorem is referenced by:  fcoi1 3630  fcoi2 3631  dmfco 3758  fvco 3759

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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