Theorem rabxp 4684

Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)

Hypothesis

Ref	Expression
rabxp.1	|- ( x = <. y , z >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabxp	|- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }

Distinct variable groups:   x, y, z, A   x, B, y, z   ph, y, z   ps, x

Allowed substitution hints:   ph( x)   ps( y, z)

Proof of Theorem rabxp

Step	Hyp	Ref	Expression
1		elxp 4664	. . . . 5 |- ( x e. ( A X. B ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) )
2	1	anbi1i 458	. . . 4 |- ( ( x e. ( A X. B ) /\ ph ) <-> ( E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) )
3		19.41vv 1915	. . . 4 |- ( E. y E. z ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> ( E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) )
4		anass 401	. . . . . 6 |- ( ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> ( x = <. y , z >. /\ ( ( y e. A /\ z e. B ) /\ ph ) ) )
5		rabxp.1	. . . . . . . . 9 |- ( x = <. y , z >. -> ( ph <-> ps ) )
6	5	anbi2d 464	. . . . . . . 8 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( ( y e. A /\ z e. B ) /\ ps ) ) )
7		df-3an 982	. . . . . . . 8 |- ( ( y e. A /\ z e. B /\ ps ) <-> ( ( y e. A /\ z e. B ) /\ ps ) )
8	6, 7	bitr4di 198	. . . . . . 7 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( y e. A /\ z e. B /\ ps ) ) )
9	8	pm5.32i 454	. . . . . 6 |- ( ( x = <. y , z >. /\ ( ( y e. A /\ z e. B ) /\ ph ) ) <-> ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
10	4, 9	bitri 184	. . . . 5 |- ( ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
11	10	2exbii 1617	. . . 4 |- ( E. y E. z ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
12	2, 3, 11	3bitr2i 208	. . 3 |- ( ( x e. ( A X. B ) /\ ph ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
13	12	abbii 2305	. 2 |- { x | ( x e. ( A X. B ) /\ ph ) } = { x | E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) }
14		df-rab 2477	. 2 |- { x e. ( A X. B ) | ph } = { x | ( x e. ( A X. B ) /\ ph ) }
15		df-opab 4083	. 2 |- { <. y , z >. | ( y e. A /\ z e. B /\ ps ) } = { x | E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) }
16	13, 14, 15	3eqtr4i 2220	1 |- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   {crab 2472   <.cop 3613   {copab 4081   X. cxp 4645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-opab 4083  df-xp 4653

This theorem is referenced by: (None)