Theorem oprabex3 3046

Description: Existence of an operation 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 group(s):   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 2488	. 2 |- (H X. H) e. V
3		moeq 1431	. . . . . 6 |- E*z z = R
4	3	mosubop 1911	. . . . 5 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
5	4	mosubop 1911	. . . 4 |- E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R))
6		anass 336	. . . . . . . 8 |- (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
7	6	bi2ex 734	. . . . . . 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 968	. . . . . . 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 151	. . . . . 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	bi2ex 734	. . . . 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	bimo 1031	. . . 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 165	. . 3 |- E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R)
13	12	a1i 7	. 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 3044	1 |- F e. V

Syntax hints:   /\ wa 196  E.wex 678  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   X. cxp 2408  {copab2 3002

This theorem is referenced by:  axaddex 4059  axmulex 4060

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-oprab 3004

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