Theorem opelcog 3290

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 2487	. . . . 5 |- (y = A -> <.y, z>. = <.A, z>.)
2	1	eleq1d 1540	. . . 4 |- (y = A -> (<.y, z>. e. (C o. D) <-> <.A, z>. e. (C o. D)))
3		breq1 2622	. . . . . 6 |- (y = A -> (yDx <-> ADx))
4	3	anbi1d 617	. . . . 5 |- (y = A -> ((yDx /\ xCz) <-> (ADx /\ xCz)))
5	4	exbidv 1279	. . . 4 |- (y = A -> (E.x(yDx /\ xCz) <-> E.x(ADx /\ xCz)))
6	2, 5	bibi12d 629	. . 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 2488	. . . . 5 |- (z = B -> <.A, z>. = <.A, B>.)
8	7	eleq1d 1540	. . . 4 |- (z = B -> (<.A, z>. e. (C o. D) <-> <.A, B>. e. (C o. D)))
9		breq2 2623	. . . . . 6 |- (z = B -> (xCz <-> xCB))
10	9	anbi2d 616	. . . . 5 |- (z = B -> ((ADx /\ xCz) <-> (ADx /\ xCB)))
11	10	exbidv 1279	. . . 4 |- (z = B -> (E.x(ADx /\ xCz) <-> E.x(ADx /\ xCB)))
12	8, 11	bibi12d 629	. . 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 1813	. . . 4 |- y e. V
14		visset 1813	. . . 4 |- z e. V
15	13, 14	opelco 3288	. . 3 |- (<.y, z>. e. (C o. D) <-> E.x(yDx /\ xCz))
16	6, 12, 15	vtocl2g 1850	. 2 |- ((A e. R /\ B e. S) -> (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB)))
17		df-br 2620	. . . 4 |- (ADx <-> <.A, x>. e. D)
18		df-br 2620	. . . 4 |- (xCB <-> <.x, B>. e. C)
19	17, 18	anbi12i 482	. . 3 |- ((ADx /\ xCB) <-> (<.A, x>. e. D /\ <.x, B>. e. C))
20	19	exbii 1051	. 2 |- (E.x(ADx /\ xCB) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C))
21	16, 20	syl6bb 536	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 956   e. wcel 958  E.wex 980  <.cop 2411   class class class wbr 2619   o. ccom 3174

This theorem is referenced by:  fcoi1 3645  fcoi2 3646  dmfco 3773  fvco 3774

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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