Theorem rabxp 4323

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 4305	. . . . 5 |- (x e. (A X. B) <-> E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B)))
2	1	anbi1i 431	. . . 4 |- ((x e. (A X. B) /\ ph) <-> (E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B)) /\ ph))
3		19.41vv 1780	. . . 4 |- (E.yE.z((x = <.y , z >. /\ (y e. A /\ z e. B)) /\ ph) <-> (E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B)) /\ ph))
4		anass 381	. . . . . 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 437	. . . . . . . 8 |- (x = <.y , z >. -> (((y e. A /\ z e. B) /\ ph) <-> ((y e. A /\ z e. B) /\ ps)))
7		df-3an 886	. . . . . . . 8 |- ((y e. A /\ z e. B /\ ps) <-> ((y e. A /\ z e. B) /\ ps))
8	6, 7	syl6bbr 187	. . . . . . 7 |- (x = <.y , z >. -> (((y e. A /\ z e. B) /\ ph) <-> (y e. A /\ z e. B /\ ps)))
9	8	pm5.32i 427	. . . . . 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 173	. . . . 5 |- (((x = <.y , z >. /\ (y e. A /\ z e. B)) /\ ph) <-> (x = <.y , z >. /\ (y e. A /\ z e. B /\ ps)))
11	10	2exbii 1494	. . . 4 |- (E.yE.z((x = <.y , z >. /\ (y e. A /\ z e. B)) /\ ph) <-> E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B /\ ps)))
12	2, 3, 11	3bitr2i 197	. . 3 |- ((x e. (A X. B) /\ ph) <-> E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B /\ ps)))
13	12	abbii 2150	. 2 |- {x | (x e. (A X. B) /\ ph) } = {x | E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B /\ ps)) }
14		df-rab 2309	. 2 |- {x e. (A X. B) | ph } = {x | (x e. (A X. B) /\ ph) }
15		df-opab 3810	. 2 |- { <.y , z >. | (y e. A /\ z e. B /\ ps) } = {x | E.yE.z(x = <.y , z >. /\ (y e. A /\ z e. B /\ ps)) }
16	13, 14, 15	3eqtr4i 2067	1 |- {x e. (A X. B) | ph } = { <.y , z >. | (y e. A /\ z e. B /\ ps) }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023   {crab 2304   <.cop 3370   {copab 3808   X. cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294

This theorem is referenced by: (None)