Theorem elrnoprab 3054

Description: Membership in the range of an operation abstraction.

Hypotheses

Ref	Expression
elrnoprab.1	|- C e. V
elrnoprab.2	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Assertion

Ref	Expression
elrnoprab	|- (D e. ran F <-> E.x e. A E.y e. B D = C)

Distinct variable group(s):   x,y,z,A   x,B,y,z   z,C   x,D,y

Proof of Theorem elrnoprab

Step	Hyp	Ref	Expression
1		elrnoprab.1	. . 3 |- C e. V
2		elrnoprab.2	. . 3 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
3	1, 2	fnoprab2 3039	. 2 |- F Fn (A X. B)
4		fvelrn 2883	. . 3 |- (F Fn (A X. B) -> (D e. ran F <-> E.w e. (A X. B)(F` w) = D))
5		fveq2 2832	. . . . . . . 8 |- (w = <.v, u>. -> (F` w) = (F` <.v, u>.))
6	5	cleq1d 1109	. . . . . . 7 |- (w = <.v, u>. -> ((F` w) = D <-> (F` <.v, u>.) = D))
7	6	cbvop 2473	. . . . . 6 |- (E.w e. (A X. B)(F` w) = D <-> E.v e. A E.u e. B (F` <.v, u>.) = D)
8		ax-17 925	. . . . . . . 8 |- (u e. B -> A.x u e. B)
9		hboprab1 3023	. . . . . . . . . . 11 |- (w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} -> A.x w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
10	2	eleq2i 1153	. . . . . . . . . . 11 |- (w e. F <-> w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
11	10	bial 695	. . . . . . . . . . 11 |- (A.x w e. F <-> A.x w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
12	9, 10, 11	3imtr4 192	. . . . . . . . . 10 |- (w e. F -> A.x w e. F)
13		ax-17 925	. . . . . . . . . 10 |- (w e. <.v, u>. -> A.x w e. <.v, u>.)
14	12, 13	hbfv 2837	. . . . . . . . 9 |- (w e. (F` <.v, u>.) -> A.x w e. (F` <.v, u>.))
15		ax-17 925	. . . . . . . . 9 |- (w e. D -> A.x w e. D)
16	14, 15	hbeq 1171	. . . . . . . 8 |- ((F` <.v, u>.) = D -> A.x(F` <.v, u>.) = D)
17	8, 16	hbrex 1238	. . . . . . 7 |- (E.u e. B (F` <.v, u>.) = D -> A.xE.u e. B (F` <.v, u>.) = D)
18		ax-17 925	. . . . . . 7 |- (E.u e. B (F` <.x, u>.) = D -> A.vE.u e. B (F` <.x, u>.) = D)
19		opeq1 1876	. . . . . . . . . 10 |- (v = x -> <.v, u>. = <.x, u>.)
20	19	fveq2d 2836	. . . . . . . . 9 |- (v = x -> (F` <.v, u>.) = (F` <.x, u>.))
21	20	cleq1d 1109	. . . . . . . 8 |- (v = x -> ((F` <.v, u>.) = D <-> (F` <.x, u>.) = D))
22	21	birexdv 1220	. . . . . . 7 |- (v = x -> (E.u e. B (F` <.v, u>.) = D <-> E.u e. B (F` <.x, u>.) = D))
23	17, 18, 22	cbvrex 1332	. . . . . 6 |- (E.v e. A E.u e. B (F` <.v, u>.) = D <-> E.x e. A E.u e. B (F` <.x, u>.) = D)
24		hboprab2 3024	. . . . . . . . . . 11 |- (w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} -> A.y w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
25	10	bial 695	. . . . . . . . . . 11 |- (A.y w e. F <-> A.y w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
26	24, 10, 25	3imtr4 192	. . . . . . . . . 10 |- (w e. F -> A.y w e. F)
27		ax-17 925	. . . . . . . . . 10 |- (w e. <.x, u>. -> A.y w e. <.x, u>.)
28	26, 27	hbfv 2837	. . . . . . . . 9 |- (w e. (F` <.x, u>.) -> A.y w e. (F` <.x, u>.))
29		ax-17 925	. . . . . . . . 9 |- (w e. D -> A.y w e. D)
30	28, 29	hbeq 1171	. . . . . . . 8 |- ((F` <.x, u>.) = D -> A.y(F` <.x, u>.) = D)
31		ax-17 925	. . . . . . . 8 |- ((F` <.x, y>.) = D -> A.u(F` <.x, y>.) = D)
32		opeq2 1877	. . . . . . . . . 10 |- (u = y -> <.x, u>. = <.x, y>.)
33	32	fveq2d 2836	. . . . . . . . 9 |- (u = y -> (F` <.x, u>.) = (F` <.x, y>.))
34	33	cleq1d 1109	. . . . . . . 8 |- (u = y -> ((F` <.x, u>.) = D <-> (F` <.x, y>.) = D))
35	30, 31, 34	cbvrex 1332	. . . . . . 7 |- (E.u e. B (F` <.x, u>.) = D <-> E.y e. B (F` <.x, y>.) = D)
36	35	birex 1224	. . . . . 6 |- (E.x e. A E.u e. B (F` <.x, u>.) = D <-> E.x e. A E.y e. B (F` <.x, y>.) = D)
37	7, 23, 36	3bitr 155	. . . . 5 |- (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B (F` <.x, y>.) = D)
38		df-opr 3003	. . . . . . 7 |- (xFy) = (F` <.x, y>.)
39	38	cleq1i 1108	. . . . . 6 |- ((xFy) = D <-> (F` <.x, y>.) = D)
40	39	bi2rex 1226	. . . . 5 |- (E.x e. A E.y e. B (xFy) = D <-> E.x e. A E.y e. B (F` <.x, y>.) = D)
41	37, 40	bitr4 154	. . . 4 |- (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B (xFy) = D)
42	2	oprabval4g 3053	. . . . . . . 8 |- ((x e. A /\ y e. B /\ C e. V) -> (xFy) = C)
43	1, 42	mp3an3 641	. . . . . . 7 |- ((x e. A /\ y e. B) -> (xFy) = C)
44	43	cleq1d 1109	. . . . . 6 |- ((x e. A /\ y e. B) -> ((xFy) = D <-> C = D))
45		cleqcom 1103	. . . . . 6 |- (C = D <-> D = C)
46	44, 45	syl6bb 414	. . . . 5 |- ((x e. A /\ y e. B) -> ((xFy) = D <-> D = C))
47	46	bi2rexa 1230	. . . 4 |- (E.x e. A E.y e. B (xFy) = D <-> E.x e. A E.y e. B D = C)
48	41, 47	bitr 151	. . 3 |- (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B D = C)
49	4, 48	syl6bb 414	. 2 |- (F Fn (A X. B) -> (D e. ran F <-> E.x e. A E.y e. B D = C))
50	3, 49	ax-mp 6	1 |- (D e. ran F <-> E.x e. A E.y e. B D = C)

Syntax hints:   <-> wb 127   /\ wa 196  A.wal 672   = weq 797   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348  <.cop 1810   X. cxp 2408  ran crn 2411   Fn wfn 2417  ` cfv 2422  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  unxpdomlem 3649  qnnen 4931

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-un 1076  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-fn 2433  df-fv 2438  df-opr 3003  df-oprab 3004

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