Theorem rabxp 4815

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 4802	. . . . 5 |- (x e. (A X. B) <-> E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)))
2	1	anbi1i 676	. . . 4 |- ((x e. (A X. B) /\ ph) <-> (E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)) /\ ph))
3		19.41vv 1902	. . . 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 630	. . . . . 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 684	. . . . . . . 8 |- (x = <.y, z>. -> (((y e. A /\ z e. B) /\ ph) <-> ((y e. A /\ z e. B) /\ ps)))
7		df-3an 936	. . . . . . . 8 |- ((y e. A /\ z e. B /\ ps) <-> ((y e. A /\ z e. B) /\ ps))
8	6, 7	syl6bbr 254	. . . . . . 7 |- (x = <.y, z>. -> (((y e. A /\ z e. B) /\ ph) <-> (y e. A /\ z e. B /\ ps)))
9	8	pm5.32i 618	. . . . . 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 240	. . . . 5 |- (((x = <.y, z>. /\ (y e. A /\ z e. B)) /\ ph) <-> (x = <.y, z>. /\ (y e. A /\ z e. B /\ ps)))
11	10	2exbii 1583	. . . 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 264	. . 3 |- ((x e. (A X. B) /\ ph) <-> E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B /\ ps)))
13	12	abbii 2466	. 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 2624	. 2 |- {x e. (A X. B) | ph} = {x | (x e. (A X. B) /\ ph)}
15		df-opab 4624	. 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 2383	1 |- {x e. (A X. B) | ph} = {<.y, z>. | (y e. A /\ z e. B /\ ps)}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2619  <.cop 4562  {copab 4623   X. cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-xp 4785

This theorem is referenced by: (None)