Theorem dfoprab2 3021

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 group(s):   x,z,w   y,z,w   ph,w

Proof of Theorem dfoprab2

Step	Hyp	Ref	Expression
1		excom 728	. . . 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 778	. . . . 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 966	. . . . . . 7 |- (E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.) <-> ((v = <.<.x, y>., z>. /\ ph) /\ E.w w = <.x, y>.))
4		opeq1 1876	. . . . . . . . . . . 12 |- (w = <.x, y>. -> <.w, z>. = <.<.x, y>., z>.)
5	4	cleq2d 1112	. . . . . . . . . . 11 |- (w = <.x, y>. -> (v = <.w, z>. <-> v = <.<.x, y>., z>.))
6	5	pm5.32ri 490	. . . . . . . . . 10 |- ((v = <.w, z>. /\ w = <.x, y>.) <-> (v = <.<.x, y>., z>. /\ w = <.x, y>.))
7	6	anbi1i 368	. . . . . . . . 9 |- (((v = <.w, z>. /\ w = <.x, y>.) /\ ph) <-> ((v = <.<.x, y>., z>. /\ w = <.x, y>.) /\ ph))
8		anass 336	. . . . . . . . 9 |- (((v = <.w, z>. /\ w = <.x, y>.) /\ ph) <-> (v = <.w, z>. /\ (w = <.x, y>. /\ ph)))
9		an23 371	. . . . . . . . 9 |- (((v = <.<.x, y>., z>. /\ w = <.x, y>.) /\ ph) <-> ((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
10	7, 8, 9	3bitr3 156	. . . . . . . 8 |- ((v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> ((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
11	10	biex 733	. . . . . . 7 |- (E.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
12		opex 1893	. . . . . . . . 9 |- <.x, y>. e. V
13	12	isseti 1352	. . . . . . . 8 |- E.w w = <.x, y>.
14	13	biantru 543	. . . . . . 7 |- ((v = <.<.x, y>., z>. /\ ph) <-> ((v = <.<.x, y>., z>. /\ ph) /\ E.w w = <.x, y>.))
15	3, 11, 14	3bitr4 158	. . . . . 6 |- (E.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> (v = <.<.x, y>., z>. /\ ph))
16	15	bi3ex 735	. . . . 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 151	. . . 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 968	. . . . 5 |- (E.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> (v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph)))
19	18	bi2ex 734	. . . 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 156	. . 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	biabi 1181	. 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 3004	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
23		df-opab 2098	. 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 1126	1 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}

Syntax hints:   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091  <.cop 1810  {copab 2055  {copab2 3002

This theorem is referenced by:  reloprab 3022  bioprabd 3025  cbvoprab3v 3030  dmoprab 3031  rnoprab 3033  ssoprab2i 3036  funoprab 3037  fnoprval 3042

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-oprab 3004

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