Theorem rabxp 4576

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 4556	. . . . 5 |- ( x e. ( A X. B ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) )
2	1	anbi1i 453	. . . 4 |- ( ( x e. ( A X. B ) /\ ph ) <-> ( E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) )
3		19.41vv 1875	. . . 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 398	. . . . . 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 459	. . . . . . . 8 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( ( y e. A /\ z e. B ) /\ ps ) ) )
7		df-3an 964	. . . . . . . 8 |- ( ( y e. A /\ z e. B /\ ps ) <-> ( ( y e. A /\ z e. B ) /\ ps ) )
8	6, 7	syl6bbr 197	. . . . . . 7 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( y e. A /\ z e. B /\ ps ) ) )
9	8	pm5.32i 449	. . . . . 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 183	. . . . 5 |- ( ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
11	10	2exbii 1585	. . . 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 207	. . 3 |- ( ( x e. ( A X. B ) /\ ph ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
13	12	abbii 2255	. 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 2425	. 2 |- { x e. ( A X. B ) | ph } = { x | ( x e. ( A X. B ) /\ ph ) }
15		df-opab 3990	. 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 2170	1 |- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   {crab 2420   <.cop 3530   {copab 3988   X. cxp 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545

This theorem is referenced by: (None)