Theorem oprabex3 4013

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 3255	. 2 |- (H X. H) e. V
3		moeq 1916	. . . . . 6 |- E*z z = R
4	3	mosubop 2800	. . . . 5 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
5	4	mosubop 2800	. . . 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 1050	. . . . . . 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 1308	. . . . . . 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 1050	. . . . 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 1403	. . . 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 4010	1 |- F e. V

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E*wmo 1379  Vcvv 1807  <.cop 2407   X. cxp 3163  {copab2 3955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-oprab 3957

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