Theorem elrnoprabg 5066

Description: Membership in the range of an operation class abstraction.

Hypothesis

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

Assertion

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

Distinct variable groups:   x,y,z,A   x,B,y,z   z,C   x,D,y

Proof of Theorem elrnoprabg

Step	Hyp	Ref	Expression
1		elisset 2299	. . . . 5 |- (C e. R -> C e. _V)
2	1	ralimi 2168	. . . 4 |- (A.y e. B C e. R -> A.y e. B C e. _V)
3	2	ralimi 2168	. . 3 |- (A.x e. A A.y e. B C e. R -> A.x e. A A.y e. B C e. _V)
4		elrnoprabg.1	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
5	4	fnoprab2g 5063	. . 3 |- (A.x e. A A.y e. B C e. _V <-> F Fn (A X. B))
6	3, 5	sylib 215	. 2 |- (A.x e. A A.y e. B C e. R -> F Fn (A X. B))
7		fvelrnb 4719	. . 3 |- (F Fn (A X. B) -> (D e. ran F <-> E.w e. (A X. B)(F` w) = D))
8		hbra1 2147	. . . . 5 |- (A.x e. A A.y e. B C e. R -> A.xA.x e. A A.y e. B C e. R)
9		ax-17 1317	. . . . . . . 8 |- (x e. A -> A.y x e. A)
10		hbra1 2147	. . . . . . . 8 |- (A.y e. B C e. R -> A.yA.y e. B C e. R)
11	9, 10	hbral 2146	. . . . . . 7 |- (A.x e. A A.y e. B C e. R -> A.yA.x e. A A.y e. B C e. R)
12	11, 9	hban 1356	. . . . . 6 |- ((A.x e. A A.y e. B C e. R /\ x e. A) -> A.y(A.x e. A A.y e. B C e. R /\ x e. A))
13		ra42 2157	. . . . . . . . 9 |- (A.x e. A A.y e. B C e. R -> ((x e. A /\ y e. B) -> C e. R))
14	4	oprabval4g 4960	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B /\ C e. R) -> (xFy) = C)
15	14	eqeq1d 1892	. . . . . . . . . . 11 |- ((x e. A /\ y e. B /\ C e. R) -> ((xFy) = D <-> C = D))
16		eqcom 1886	. . . . . . . . . . 11 |- (C = D <-> D = C)
17	15, 16	syl6bb 595	. . . . . . . . . 10 |- ((x e. A /\ y e. B /\ C e. R) -> ((xFy) = D <-> D = C))
18	17	3expia 1069	. . . . . . . . 9 |- ((x e. A /\ y e. B) -> (C e. R -> ((xFy) = D <-> D = C)))
19	13, 18	sylcom 62	. . . . . . . 8 |- (A.x e. A A.y e. B C e. R -> ((x e. A /\ y e. B) -> ((xFy) = D <-> D = C)))
20	19	exp3a 405	. . . . . . 7 |- (A.x e. A A.y e. B C e. R -> (x e. A -> (y e. B -> ((xFy) = D <-> D = C))))
21	20	imp31 389	. . . . . 6 |- (((A.x e. A A.y e. B C e. R /\ x e. A) /\ y e. B) -> ((xFy) = D <-> D = C))
22	12, 21	rexbida 2118	. . . . 5 |- ((A.x e. A A.y e. B C e. R /\ x e. A) -> (E.y e. B (xFy) = D <-> E.y e. B D = C))
23	8, 22	rexbida 2118	. . . 4 |- (A.x e. A A.y e. B C e. R -> (E.x e. A E.y e. B (xFy) = D <-> E.x e. A E.y e. B D = C))
24		fveq2 4681	. . . . . . . 8 |- (w = <.v, u>. -> (F` w) = (F` <.v, u>.))
25		df-opr 4886	. . . . . . . 8 |- (vFu) = (F` <.v, u>.)
26	24, 25	syl6eqr 1946	. . . . . . 7 |- (w = <.v, u>. -> (F` w) = (vFu))
27	26	eqeq1d 1892	. . . . . 6 |- (w = <.v, u>. -> ((F` w) = D <-> (vFu) = D))
28	27	rexxp 4042	. . . . 5 |- (E.w e. (A X. B)(F` w) = D <-> E.v e. A E.u e. B (vFu) = D)
29		ax-17 1317	. . . . . . 7 |- (u e. B -> A.x u e. B)
30		ax-17 1317	. . . . . . . . 9 |- (w e. v -> A.x w e. v)
31		hboprab1 4919	. . . . . . . . . 10 |- (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)})
32	4, 31	hbxfr 1992	. . . . . . . . 9 |- (w e. F -> A.x w e. F)
33		ax-17 1317	. . . . . . . . 9 |- (w e. u -> A.x w e. u)
34	30, 32, 33	hbopr 4904	. . . . . . . 8 |- (w e. (vFu) -> A.x w e. (vFu))
35		ax-17 1317	. . . . . . . 8 |- (w e. D -> A.x w e. D)
36	34, 35	hbeq 1995	. . . . . . 7 |- ((vFu) = D -> A.x(vFu) = D)
37	29, 36	hbrex 2149	. . . . . 6 |- (E.u e. B (vFu) = D -> A.xE.u e. B (vFu) = D)
38		ax-17 1317	. . . . . 6 |- (E.u e. B (xFu) = D -> A.vE.u e. B (xFu) = D)
39		opreq1 4889	. . . . . . . 8 |- (v = x -> (vFu) = (xFu))
40	39	eqeq1d 1892	. . . . . . 7 |- (v = x -> ((vFu) = D <-> (xFu) = D))
41	40	rexbidv 2124	. . . . . 6 |- (v = x -> (E.u e. B (vFu) = D <-> E.u e. B (xFu) = D))
42	37, 38, 41	cbvrex 2279	. . . . 5 |- (E.v e. A E.u e. B (vFu) = D <-> E.x e. A E.u e. B (xFu) = D)
43		ax-17 1317	. . . . . . . . 9 |- (w e. x -> A.y w e. x)
44		hboprab2 4920	. . . . . . . . . 10 |- (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)})
45	4, 44	hbxfr 1992	. . . . . . . . 9 |- (w e. F -> A.y w e. F)
46		ax-17 1317	. . . . . . . . 9 |- (w e. u -> A.y w e. u)
47	43, 45, 46	hbopr 4904	. . . . . . . 8 |- (w e. (xFu) -> A.y w e. (xFu))
48		ax-17 1317	. . . . . . . 8 |- (w e. D -> A.y w e. D)
49	47, 48	hbeq 1995	. . . . . . 7 |- ((xFu) = D -> A.y(xFu) = D)
50		ax-17 1317	. . . . . . 7 |- ((xFy) = D -> A.u(xFy) = D)
51		opreq2 4890	. . . . . . . 8 |- (u = y -> (xFu) = (xFy))
52	51	eqeq1d 1892	. . . . . . 7 |- (u = y -> ((xFu) = D <-> (xFy) = D))
53	49, 50, 52	cbvrex 2279	. . . . . 6 |- (E.u e. B (xFu) = D <-> E.y e. B (xFy) = D)
54	53	rexbii 2128	. . . . 5 |- (E.x e. A E.u e. B (xFu) = D <-> E.x e. A E.y e. B (xFy) = D)
55	28, 42, 54	3bitri 194	. . . 4 |- (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B (xFy) = D)
56	23, 55	syl5bb 591	. . 3 |- (A.x e. A A.y e. B C e. R -> (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B D = C))
57	7, 56	sylan9bbr 600	. 2 |- ((A.x e. A A.y e. B C e. R /\ F Fn (A X. B)) -> (D e. ran F <-> E.x e. A E.y e. B D = C))
58	6, 57	mpdan 768	1 |- (A.x e. A A.y e. B C e. R -> (D e. ran F <-> E.x e. A E.y e. B D = C))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292  <.cop 3046   X. cxp 3984  ran crn 3987   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  {copab2 4885

This theorem is referenced by:  elrnoprab 5067  blrn 9118

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021

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