Theorem ov3 5599

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ov3.1	|- S e. _V
ov3.2	|- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)
ov3.3	|- 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
ov3	|- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)

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

Allowed substitution hints:   C(x)   D(x)   R(w,v,u,f)   S(x,y)   F(x,y,z,w,v,u,f)

Proof of Theorem ov3

Step	Hyp	Ref	Expression
1		ov3.1	. . 3 |- S e. _V
2	1	isseti 2865	. 2 |- E.z z = S
3		nfv 1619	. . 3 |- F/z((A e. H /\ B e. H) /\ (C e. H /\ D e. H))
4		nfcv 2489	. . . . 5 |- F/_z<.A, B>.
5		ov3.3	. . . . . 6 |- 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))}
6		nfoprab3 5548	. . . . . 6 |- F/_z{<.<.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))}
7	5, 6	nfcxfr 2486	. . . . 5 |- F/_zF
8		nfcv 2489	. . . . 5 |- F/_z<.C, D>.
9	4, 7, 8	nfov 5545	. . . 4 |- F/_z(<.A, B>.F<.C, D>.)
10	9	nfeq1 2498	. . 3 |- F/z(<.A, B>.F<.C, D>.) = S
11		ov3.2	. . . . . . 7 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)
12	11	eqeq2d 2364	. . . . . 6 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> (z = R <-> z = S))
13	12	copsex4g 4610	. . . . 5 |- (((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>.) /\ z = R) <-> z = S))
14		opelxp 4811	. . . . . 6 |- (<.A, B>. e. (H X. H) <-> (A e. H /\ B e. H))
15		opelxp 4811	. . . . . 6 |- (<.C, D>. e. (H X. H) <-> (C e. H /\ D e. H))
16		nfcv 2489	. . . . . . 7 |- F/_x<.A, B>.
17		nfcv 2489	. . . . . . 7 |- F/_y<.A, B>.
18		nfcv 2489	. . . . . . 7 |- F/_y<.C, D>.
19		nfv 1619	. . . . . . . 8 |- F/xE.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R)
20		nfoprab1 5546	. . . . . . . . . . 11 |- F/_x{<.<.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))}
21	5, 20	nfcxfr 2486	. . . . . . . . . 10 |- F/_xF
22		nfcv 2489	. . . . . . . . . 10 |- F/_xy
23	16, 21, 22	nfov 5545	. . . . . . . . 9 |- F/_x(<.A, B>.Fy)
24	23	nfeq1 2498	. . . . . . . 8 |- F/x(<.A, B>.Fy) = z
25	19, 24	nfim 1813	. . . . . . 7 |- F/x(E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) -> (<.A, B>.Fy) = z)
26		nfv 1619	. . . . . . . 8 |- F/yE.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R)
27		nfoprab2 5547	. . . . . . . . . . 11 |- F/_y{<.<.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))}
28	5, 27	nfcxfr 2486	. . . . . . . . . 10 |- F/_yF
29	17, 28, 18	nfov 5545	. . . . . . . . 9 |- F/_y(<.A, B>.F<.C, D>.)
30	29	nfeq1 2498	. . . . . . . 8 |- F/y(<.A, B>.F<.C, D>.) = z
31	26, 30	nfim 1813	. . . . . . 7 |- F/y(E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R) -> (<.A, B>.F<.C, D>.) = z)
32		eqeq1 2359	. . . . . . . . . . 11 |- (x = <.A, B>. -> (x = <.w, v>. <-> <.A, B>. = <.w, v>.))
33	32	anbi1d 685	. . . . . . . . . 10 |- (x = <.A, B>. -> ((x = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ y = <.u, f>.)))
34	33	anbi1d 685	. . . . . . . . 9 |- (x = <.A, B>. -> (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R)))
35	34	4exbidv 1630	. . . . . . . 8 |- (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)))
36		oveq1 5530	. . . . . . . . 9 |- (x = <.A, B>. -> (xFy) = (<.A, B>.Fy))
37	36	eqeq1d 2361	. . . . . . . 8 |- (x = <.A, B>. -> ((xFy) = z <-> (<.A, B>.Fy) = z))
38	35, 37	imbi12d 311	. . . . . . 7 |- (x = <.A, B>. -> ((E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) -> (xFy) = z) <-> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) -> (<.A, B>.Fy) = z)))
39		eqeq1 2359	. . . . . . . . . . 11 |- (y = <.C, D>. -> (y = <.u, f>. <-> <.C, D>. = <.u, f>.))
40	39	anbi2d 684	. . . . . . . . . 10 |- (y = <.C, D>. -> ((<.A, B>. = <.w, v>. /\ y = <.u, f>.) <-> (<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.)))
41	40	anbi1d 685	. . . . . . . . 9 |- (y = <.C, D>. -> (((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> ((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R)))
42	41	4exbidv 1630	. . . . . . . 8 |- (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)))
43		oveq2 5531	. . . . . . . . 9 |- (y = <.C, D>. -> (<.A, B>.Fy) = (<.A, B>.F<.C, D>.))
44	43	eqeq1d 2361	. . . . . . . 8 |- (y = <.C, D>. -> ((<.A, B>.Fy) = z <-> (<.A, B>.F<.C, D>.) = z))
45	42, 44	imbi12d 311	. . . . . . 7 |- (y = <.C, D>. -> ((E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ y = <.u, f>.) /\ z = R) -> (<.A, B>.Fy) = z) <-> (E.wE.vE.uE.f((<.A, B>. = <.w, v>. /\ <.C, D>. = <.u, f>.) /\ z = R) -> (<.A, B>.F<.C, D>.) = z)))
46		moeq 3012	. . . . . . . . . . . 12 |- E*z z = R
47	46	mosubop 4613	. . . . . . . . . . 11 |- E*zE.uE.f(y = <.u, f>. /\ z = R)
48	47	mosubop 4613	. . . . . . . . . 10 |- E*zE.wE.v(x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R))
49		anass 630	. . . . . . . . . . . . . 14 |- (((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
50	49	2exbii 1583	. . . . . . . . . . . . 13 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)))
51		19.42vv 1907	. . . . . . . . . . . . 13 |- (E.uE.f(x = <.w, v>. /\ (y = <.u, f>. /\ z = R)) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
52	50, 51	bitri 240	. . . . . . . . . . . 12 |- (E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) <-> (x = <.w, v>. /\ E.uE.f(y = <.u, f>. /\ z = R)))
53	52	2exbii 1583	. . . . . . . . . . 11 |- (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)))
54	53	mobii 2240	. . . . . . . . . 10 |- (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)))
55	48, 54	mpbir 200	. . . . . . . . 9 |- E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R)
56	55	a1i 10	. . . . . . . 8 |- ((x e. (H X. H) /\ y e. (H X. H)) -> E*zE.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))
57	56, 5	ovidi 5594	. . . . . . 7 |- ((x e. (H X. H) /\ y e. (H X. H)) -> (E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R) -> (xFy) = z))
58	16, 17, 18, 25, 31, 38, 45, 57	vtocl2gaf 2921	. . . . . 6 |- ((<.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>.) /\ z = R) -> (<.A, B>.F<.C, D>.) = z))
59	14, 15, 58	syl2anbr 466	. . . . 5 |- (((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>.) /\ z = R) -> (<.A, B>.F<.C, D>.) = z))
60	13, 59	sylbird 226	. . . 4 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (z = S -> (<.A, B>.F<.C, D>.) = z))
61		eqeq2 2362	. . . 4 |- (z = S -> ((<.A, B>.F<.C, D>.) = z <-> (<.A, B>.F<.C, D>.) = S))
62	60, 61	mpbidi 207	. . 3 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (z = S -> (<.A, B>.F<.C, D>.) = S))
63	3, 10, 62	exlimd 1806	. 2 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (E.z z = S -> (<.A, B>.F<.C, D>.) = S))
64	2, 63	mpi 16	1 |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E*wmo 2205  _Vcvv 2859  <.cop 4561   X. cxp 4770  (class class class)co 5525  {coprab 5527

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528

This theorem is referenced by: (None)