Theorem dfoprab2 3991

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 1046	. . . 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 1100	. . . . 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 1308	. . . . . . 7 |- (E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.) <-> ((v = <.<.x, y>., z>. /\ ph) /\ E.w w = <.x, y>.))
4		opeq1 2487	. . . . . . . . . . . 12 |- (w = <.x, y>. -> <.w, z>. = <.<.x, y>., z>.)
5	4	eqeq2d 1486	. . . . . . . . . . 11 |- (w = <.x, y>. -> (v = <.w, z>. <-> v = <.<.x, y>., z>.))
6	5	pm5.32ri 646	. . . . . . . . . 10 |- ((v = <.w, z>. /\ w = <.x, y>.) <-> (v = <.<.x, y>., z>. /\ w = <.x, y>.))
7	6	anbi1i 481	. . . . . . . . 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 485	. . . . . . . . 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 1051	. . . . . . 7 |- (E.w(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> E.w((v = <.<.x, y>., z>. /\ ph) /\ w = <.x, y>.))
12		opex 2782	. . . . . . . . 9 |- <.x, y>. e. V
13	12	isseti 1815	. . . . . . . 8 |- E.w w = <.x, y>.
14	13	biantru 724	. . . . . . 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 1053	. . . . 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 1310	. . . . 5 |- (E.xE.y(v = <.w, z>. /\ (w = <.x, y>. /\ ph)) <-> (v = <.w, z>. /\ E.xE.y(w = <.x, y>. /\ ph)))
19	18	2exbii 1052	. . . 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 1575	. 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 3966	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
23		df-opab 2667	. 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 1505	1 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}

Syntax hints:   /\ wa 223   = wceq 956  E.wex 980  {cab 1463  <.cop 2411  {copab 2666  {copab2 3964

This theorem is referenced by:  reloprab 3992  oprabbid 3995  cbvoprab3v 4000  dmoprab 4002  rnoprab 4004  ssoprab2i 4008  resoprab 4009  funoprabg 4010  fnoprval 4017  oprabval6g 4032  dfoprab3 4114  nvvcop 8213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-oprab 3966

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