Theorem oprabex3 4007

Description: Existence of an operation class abstraction (special case).

Hypotheses

Ref	Expression
oprabex3.1	|- H e. V
oprabex3.2	|- F = {<.<.x, y>., z>. | ((x e. (H X. H) /\ y e. (H X. H)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))}

Assertion

Ref	Expression
oprabex3	|- F e. V

Distinct variable groups:   x,y,z,w,v,u,f,H   x,R,y,z

Proof of Theorem oprabex3

Step	Hyp	Ref	Expression
1		oprabex3.1	. . 3 |- H e. V
2	1, 1	xpex 3250	. 2 |- (H X. H) e. V
3		moeq 1911	. . . . . 6 |- E*z z = R
4	3	mosubop 2794	. . . . 5 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
5	4	mosubop 2794	. . . 4 |- E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R))
6		anass 439	. . . . . . . 8 |- (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
7	6	2exbii 1048	. . . . . . 7 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
8		19.42vv 1305	. . . . . . 7 |- (E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
9	7, 8	bitr 173	. . . . . 6 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
10	9	2exbii 1048	. . . . 5 |- (E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
11	10	mobii 1398	. . . 4 |- (E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
12	5, 11	mpbir 190	. . 3 |- E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R)
13	12	a1i 8	. 2 |- ((x e. (H X. H) /\ y e. (H X. H)) -> E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))
14		oprabex3.2	. 2 |- F = {<.<.x, y>., z>. | ((x e. (H X. H) /\ y e. (H X. H)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))}
15	2, 2, 13, 14	oprabex 4004	1 |- F e. V

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E*wmo 1374  Vcvv 1802  <.cop 2401   X. cxp 3158  {copab2 3949

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-9 962  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-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

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-rex 1642  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-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-oprab 3951

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