Theorem rabxp 4398

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 4380	. . . . 5 |- ( x e. ( A X. B ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) )
2	1	anbi1i 445	. . . 4 |- ( ( x e. ( A X. B ) /\ ph ) <-> ( E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) )
3		19.41vv 1824	. . . 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 393	. . . . . 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 451	. . . . . . . 8 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( ( y e. A /\ z e. B ) /\ ps ) ) )
7		df-3an 921	. . . . . . . 8 |- ( ( y e. A /\ z e. B /\ ps ) <-> ( ( y e. A /\ z e. B ) /\ ps ) )
8	6, 7	syl6bbr 196	. . . . . . 7 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( y e. A /\ z e. B /\ ps ) ) )
9	8	pm5.32i 441	. . . . . 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 182	. . . . 5 |- ( ( ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) <-> ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
11	10	2exbii 1537	. . . 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 206	. . 3 |- ( ( x e. ( A X. B ) /\ ph ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B /\ ps ) ) )
13	12	abbii 2194	. 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 2357	. 2 |- { x e. ( A X. B ) | ph } = { x | ( x e. ( A X. B ) /\ ph ) }
15		df-opab 3840	. 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 2111	1 |- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   {crab 2352   <.cop 3401   {copab 3838   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369

This theorem is referenced by: (None)