Theorem dfoprab2 3976

Description: Class abstraction for operations in terms of class abstraction of ordered pairs.

Assertion

Ref	Expression
dfoprab2	|- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}

Distinct variable groups:   x,z,w   y,z,w   ph,w

Proof of Theorem dfoprab2

Step	Hyp	Ref	Expression
1		excom 1042	. . . 4 |- (E.zE.wE.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.wE.zE.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)))
2		exrot4 1096	. . . . 5 |- (E.zE.wE.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.xE.yE.zE.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)))
3		19.42v 1303	. . . . . . 7 |- (E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.) <-> ((v = <.<.x, y>., z>. /\ ph) /\ E.w w = <.x, y>.))
4		opeq1 2478	. . . . . . . . . . . 12 |- (w = <.x, y>. -> <.w, z>. = <.<.x, y>., z>.)
5	4	eqeq2d 1478	. . . . . . . . . . 11 |- (w = <.x, y>. -> (v = <.w, z>. <-> v = <.<.x, y>., z>.))
6	5	pm5.32ri 644	. . . . . . . . . 10 |- ((v = <.w, z>. /\ w = <.x, y>.) <-> (v = <.<.x, y>., z>. /\ w = <.x, y>.))
7	6	anbi1i 480	. . . . . . . . 9 |- (((v = <.w, z>. /\ w = <.x, y>.) /\ ph) <-> ((v = <.<.x, y>., z>. /\ w = <.x, y>.) /\ ph))
8		anass 439	. . . . . . . . 9 |- (((v = <.w, z>. /\ w = <.x, y>.) /\ ph) <-> (v = <.w, z>. /\ (w = <.x, y>. /\ ph)))
9		an23 484	. . . . . . . . 9 |- (((v = <.<.x, y>., z>. /\ w = <.x, y>.) /\ ph) <-> ((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
10	7, 8, 9	3bitr3 181	. . . . . . . 8 |- ((v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> ((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
11	10	exbii 1047	. . . . . . 7 |- (E.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
12		opex 2772	. . . . . . . . 9 |- <.x, y>. e. V
13	12	isseti 1806	. . . . . . . 8 |- E.w w = <.x, y>.
14	13	biantru 722	. . . . . . 7 |- ((v = <.<.x, y>., z>. /\ ph) <-> ((v = <.<.x, y>., z>. /\ ph) /\ E.w w = <.x, y>.))
15	3, 11, 14	3bitr4 183	. . . . . 6 |- (E.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> (v = <.<.x, y>., z>. /\ ph))
16	15	3exbi 1049	. . . . 5 |- (E.xE.yE.zE.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.xE.yE.z(v = <.<.x, y>., z>. /\ ph))
17	2, 16	bitr 173	. . . 4 |- (E.zE.wE.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.xE.yE.z(v = <.<.x, y>., z>. /\ ph))
18		19.42vv 1305	. . . . 5 |- (E.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> (v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph)))
19	18	2exbii 1048	. . . 4 |- (E.wE.zE.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.wE.z(v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph)))
20	1, 17, 19	3bitr3 181	. . 3 |- (E.xE.yE.z(v = <.<.x, y>., z>. /\ ph) <-> E.wE.z(v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph)))
21	20	abbii 1567	. 2 |- {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)} = {v | E.wE.z(v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph))}
22		df-oprab 3951	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
23		df-opab 2657	. 2 |- {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)} = {v | E.wE.z(v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph))}
24	21, 22, 23	3eqtr4 1497	1 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}

Syntax hints:   /\ wa 223   = wceq 953  E.wex 977  {cab 1456  <.cop 2401  {copab 2656  {copab2 3949

This theorem is referenced by:  reloprab 3977  oprabbid 3980  cbvoprab3v 3985  dmoprab 3987  rnoprab 3989  ssoprab2i 3993  resoprab 3994  funoprabg 3995  fnoprval 4002  oprabval6g 4017  dfoprab3 4098  nvvcop 8151

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-oprab 3951

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