Theorem oprabval3 4030

Description: The value of an operation class abstraction. Special case.

Hypotheses

Ref	Expression
oprabval3.1	|- S e. V
oprabval3.2	|- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)
oprabval3.3	|- 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
oprabval3	|- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)

Distinct variable groups:   x,y,z,w,v,u,f,A   x,B,y,z,w,v,u,f   x,C,y,z,w,v,u,f   x,D,y,z,w,v,u,f   x,H,y,z,w,v,u,f   x,R,y,z   x,S,y,z,w,v,u,f

Proof of Theorem oprabval3

Step	Hyp	Ref	Expression
1		oprabval3.1	. . 3 |- S e. V
2		eqeq1 1481	. . . . . 6 |- (x = <.A, B>. -> (x = <.w, v>. <-> <.A, B>. = <.w, v>.))
3	2	anbi1d 617	. . . . 5 |- (x = <.A, B>. -> ((x = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ y = <.u, f>.)))
4	3	anbi1d 617	. . . 4 |- (x = <.A, B>. -> (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R)))
5	4	4exbidv 1283	. . 3 |- (x = <.A, B>. -> (E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R)))
6		eqeq1 1481	. . . . . 6 |- (y = <.C, D>. -> (y = <.u, f>. <-> <.C, D>. = <.u, f>.))
7	6	anbi2d 616	. . . . 5 |- (y = <.C, D>. -> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.)))
8	7	anbi1d 617	. . . 4 |- (y = <.C, D>. -> (((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R)))
9	8	4exbidv 1283	. . 3 |- (y = <.C, D>. -> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R)))
10		eqeq1 1481	. . . . 5 |- (z = S -> (z = R <-> S = R))
11	10	anbi2d 616	. . . 4 |- (z = S -> (((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ S = R)))
12	11	4exbidv 1283	. . 3 |- (z = S -> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R) <-> E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ S = R)))
13		moeq 1920	. . . . . . 7 |- E*z z = R
14	13	mosubop 2805	. . . . . 6 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
15	14	mosubop 2805	. . . . 5 |- E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R))
16		anass 439	. . . . . . . . 9 |- (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
17	16	2exbii 1052	. . . . . . . 8 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
18		19.42vv 1310	. . . . . . . 8 |- (E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
19	17, 18	bitr 173	. . . . . . 7 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
20	19	2exbii 1052	. . . . . 6 |- (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)))
21	20	mobii 1405	. . . . 5 |- (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)))
22	15, 21	mpbir 190	. . . 4 |- E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R)
23	22	a1i 8	. . 3 |- ((x e. (H X. H) /\ y e. (H X. H)) -> E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))
24		oprabval3.3	. . 3 |- 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))}
25	1, 5, 9, 12, 23, 24	oprabvali 4025	. 2 |- ((<.A, B>. e. (H X. H) /\ <.C, D>. e. (H X. H)) -> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ S = R) -> (<.A, B>.F<.C, D>.) = S))
26		opelxpi 3217	. . 3 |- ((A e. H /\ B e. H) -> <.A, B>. e. (H X. H))
27		opelxpi 3217	. . 3 |- ((C e. H /\ D e. H) -> <.C, D>. e. (H X. H))
28	26, 27	anim12i 333	. 2 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>. e. (H X. H) /\ <.C, D>. e. (H X. H)))
29		eqid 1475	. . 3 |- S = S
30		oprabval3.2	. . . . 5 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)
31	30	eqeq2d 1486	. . . 4 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> (S = R <-> S = S))
32	31	copsex4g 2794	. . 3 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ S = R) <-> S = S))
33	29, 32	mpbiri 194	. 2 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ S = R))
34	25, 28, 33	sylc 68	1 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E*wmo 1381  Vcvv 1811  <.cop 2411   X. cxp 3168  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  oprec 4318  addcnsr 5253  mulcnsr 5254

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-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-fv 3198  df-opr 3965  df-oprab 3966

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