Theorem optocl 3225

Description: Implicit substitution of class for ordered pair.

Hypotheses

Ref	Expression
optocl.1	|- D = (B X. C)
optocl.2	|- (<.x, y>. = A -> (ph <-> ps))
optocl.3	|- ((x e. B /\ y e. C) -> ph)

Assertion

Ref	Expression
optocl	|- (A e. D -> ps)

Distinct variable groups:   x,y,A   x,B,y   x,C,y   ps,x,y

Proof of Theorem optocl

Step	Hyp	Ref	Expression
1		optocl.1	. . 3 |- D = (B X. C)
2	1	eleq2i 1530	. 2 |- (A e. D <-> A e. (B X. C))
3		elxp3 3214	. . 3 |- (A e. (B X. C) <-> E.xE.y(<.x, y>. = A /\ <.x, y>. e. (B X. C)))
4		optocl.2	. . . . . 6 |- (<.x, y>. = A -> (ph <-> ps))
5		visset 1804	. . . . . . . 8 |- y e. V
6	5	opelxp 3204	. . . . . . 7 |- (<.x, y>. e. (B X. C) <-> (x e. B /\ y e. C))
7		optocl.3	. . . . . . 7 |- ((x e. B /\ y e. C) -> ph)
8	6, 7	sylbi 199	. . . . . 6 |- (<.x, y>. e. (B X. C) -> ph)
9	4, 8	syl5bi 208	. . . . 5 |- (<.x, y>. = A -> (<.x, y>. e. (B X. C) -> ps))
10	9	imp 350	. . . 4 |- ((<.x, y>. = A /\ <.x, y>. e. (B X. C)) -> ps)
11	10	19.23aivv 1291	. . 3 |- (E.xE.y(<.x, y>. = A /\ <.x, y>. e. (B X. C)) -> ps)
12	3, 11	sylbi 199	. 2 |- (A e. (B X. C) -> ps)
13	2, 12	sylbi 199	1 |- (A e. D -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  <.cop 2401   X. cxp 3158

This theorem is referenced by:  2optocl 3226  3optocl 3227  ecoptocl 4287  ax0id 5253  ax1id 5254  axcnre 5258

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-opab 2657  df-xp 3174

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