Theorem rabxp 4463

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 4445	. . . . 5 |- ( x e. ( A X. B ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) )
2	1	anbi1i 446	. . . 4 |- ( ( x e. ( A X. B ) /\ ph ) <-> ( E. y E. z ( x = <. y , z >. /\ ( y e. A /\ z e. B ) ) /\ ph ) )
3		19.41vv 1831	. . . 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 452	. . . . . . . 8 |- ( x = <. y , z >. -> ( ( ( y e. A /\ z e. B ) /\ ph ) <-> ( ( y e. A /\ z e. B ) /\ ps ) ) )
7		df-3an 926	. . . . . . . 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 442	. . . . . 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 1542	. . . 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 2203	. 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 2368	. 2 |- { x e. ( A X. B ) | ph } = { x | ( x e. ( A X. B ) /\ ph ) }
15		df-opab 3892	. 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 2118	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 924   = wceq 1289   E.wex 1426   e. wcel 1438   {cab 2074   {crab 2363   <.cop 3444   {copab 3890   X. cxp 4426

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434

This theorem is referenced by: (None)