Theorem oprabex3 4022

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 3260	. 2 |- (H X. H) e. V
3		moeq 1920	. . . . . 6 |- E*z z = R
4	3	mosubop 2805	. . . . 5 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
5	4	mosubop 2805	. . . 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 1052	. . . . . . 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 1310	. . . . . . 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 1052	. . . . 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 1405	. . . 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 4019	1 |- F e. V

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E*wmo 1381  Vcvv 1811  <.cop 2411   X. cxp 3168  {copab2 3964

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-9 965  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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

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-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-oprab 3966

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