Theorem oprabval3 3052

Description: The value of an operation 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 group(s):   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		cleq1 1107	. . . . . 6 |- (x = <.A, B>. -> (x = <.w, v>. <-> <.A, B>. = <.w, v>.))
3	2	anbi1d 469	. . . . 5 |- (x = <.A, B>. -> ((x = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ y = <.u, f>.)))
4	3	anbi1d 469	. . . 4 |- (x = <.A, B>. -> (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R)))
5	4	bi4exdv 940	. . 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		cleq1 1107	. . . . . 6 |- (y = <.C, D>. -> (y = <.u, f>. <-> <.C, D>. = <.u, f>.))
7	6	anbi2d 468	. . . . 5 |- (y = <.C, D>. -> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.)))
8	7	anbi1d 469	. . . 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	bi4exdv 940	. . 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		cleq1 1107	. . . . 5 |- (z = S -> (z = R <-> S = R))
11	10	anbi2d 468	. . . 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	bi4exdv 940	. . 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 1431	. . . . . . 7 |- E*z z = R
14	13	mosubop 1911	. . . . . 6 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
15	14	mosubop 1911	. . . . 5 |- E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R))
16		anass 336	. . . . . . . . 9 |- (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
17	16	bi2ex 734	. . . . . . . 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 968	. . . . . . . 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 151	. . . . . . 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	bi2ex 734	. . . . . 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	bimo 1031	. . . . 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 165	. . . 4 |- E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R)
23	22	a1i 7	. . 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 3049	. 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 2455	. . 3 |- ((A e. H /\ B e. H) -> <.A, B>. e. (H X. H))
27		opelxpi 2455	. . 3 |- ((C e. H /\ D e. H) -> <.C, D>. e. (H X. H))
28	26, 27	anim12i 268	. 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		cleqid 1102	. . 3 |- S = S
30		oprabval3.2	. . . . 5 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)
31	30	cleq2d 1112	. . . 4 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> (S = R <-> S = S))
32	31	copsex4g 1904	. . 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 169	. 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 62	1 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   X. cxp 2408  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  oprec 3254  addcnsr 4047  mulcnsr 4048

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  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-fv 2438  df-opr 3003  df-oprab 3004

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